The  D.  Van  Noftrand  Company 

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of  reduction. 


I  THE    THEORY 

OF 

ENGINEERING  DEAWING 


BY 


ALPHONSE  A.  ADLER,  B.S.,  M.E. 

Member  American  Society  of  Mechanical  Engineers;  Instructor  in  Mechanical 
Drawing  and  Designing,  Polytechnic  Institute,  Brooklyn,  N.  Y. 


NEW  YORK 

D.  VAN  NOSTRAND  COMPANY 

25   PAEK   PLACE 

1912 


Copyright,  1912 

BY 

D.  VAN  NOSTRAND  COMPANY 


THE  SCIENTIFIC  PRESS 
DNUMMOND  AND 

BROOKLYN,  N.  Y, 


PREFACE 


ALTHOUGH  the  subject  matter  of  this  volume  is,  in  large 
measure,  identical  with  that  of  many  treatises  on  descriptive 
geometry,  the  author  has  called  it  "  Theory  of  Engineering 
Drawing,"  believing  that  this  title  indicates  better  than  could 
any  other,  the  ultimate  purpose  of  the  book.  That  texts  on 
descriptive  geometry  appear  with  some  degree  of  frequency, 
with  but  few,  if  any,  additions  to  the  theory,  indicates  that 
teachers  are  aware  of  certain  weaknesses  in  existing  methods  of 
presenting  the  subject.  It  is  precisely  these  weakness  that  the 
present  work  aims  to  correct. 

The  author  emphasizes  the  fact  that  the  student  is  concerned 
with  the  representation  on  a  plane  of  objects  in  space  of  three 
dimensions.  The  analysis,  important  as  it  is,  has  for  its  primary 
purpose  the  development  of  methods  for  such  representation 
and  the  interpretation  of  the  resulting  drawings.  It  is  nowhere 
regarded  as  an  end  in  itself.  The  number  of  fundamental  prin- 
ciples has  been  reduced  to  a  minimum;  indeed  it  will  be  found 
that  the  entire  text  is  based  on  the  problem  of  finding  the  piercing 
point  of  a  given  line  on  a  given  surface,  and  a  few  additional 
operations.  The  accepted  method  of  presenting  the  subject, 
is  to  start  with  a  set  of  definitions, to  consider  in  detail  the  ortho- 
graphic projection  of  a  point,  and  then,  on  the  foundation  thus 
laid,  to  build  the  theory  of  the  projection  of  lines,  surfaces,  and 
solids.  Logical  and  beautiful  as  this  systematic  developmeen 
may  be,  it  nevertheless  presents  certain  inherent  difficulties,  chief 
of  which  is  that  the  student  is  confronted  at  the  outset  with  that 
most  abstract  of  all  abstractions,  the  mathematical  point.  In 
this  volume  the  order  of  presentation  is  reversed  and  the  reader 
is  asked  to  consider  first  some  concrete  object,  a  box,  for  instance, 
the  study  of  which  furnishes  material  of  use  in  the  later  discussion 
of  its  bounding  surfaces  and  lines. 

The  "  Theory  of  Engineering  Drawing  "  is  divided  into  four 

iii 

^59666 


iv  PREFACE 

parts.  Part  I  treats  of  oblique  projection,  orthographic  projec- 
tion, and  a  special  case  of  the  latter,  axonometric  projection.  The 
student  is  advised  to  give  special  attention  to  the  classification 
at  the  end  of  this  section,  because  it  gives  a  complete  outline  of 
the  entire  subject.  Part  II  contains  a  variety  of  problems  of 
such  nature  as  to  be  easily  understood  by  those  whose  training 
has  not  extended  to  the  more  highly  specialized  branches  of  com- 
mercial or  engineering  practice.  Part  III  considers  convergent 
projective  line  drawing,  more  familiar  under  the  name  of  perspec- 
tive. Part  IV  has  to  do  with  the  pictorial  effects  of  illumination, 
since  a  knowledge  of  shades  and  shadows  is  frequently  required 
in  the  preparation  of  complicated  drawings. 

No  claim  is  made  to  originality  of  subject  matter,  but  it  is 
not  possible  to  acknowledge  indebtedness  to  individual  writers, 
for  the  topics  discussed  have  been  widely  studied,  and  an  historical 
review  is  here  out  of  place.  The  author  wishes,  however,  to  express 
his  sense  of  obligation  to  Professor  William  J.  Berry  of  the  Depart- 
ment of  Mathematics  in  the  Polytechnic  Institute  of  Brooklyn 
for  his  criticism  of  Chapters  IX  and  X,  and  other  assistance, 
and  to  Mr.  Ernest  J.  Streubel,  M.A.,  of  the  Department  of  English 
for  his  untiring  efforts  in  preparing  the  manuscript  for  the  press. 

POLYTECHNIC  INSTITUTE  OF  BROOKLYN, 
October,  1912. 


CONTENTS 


PART  I 

THE  PRINCIPLES  OF  PARALLEL  PROJECT  ING-LINE 

DRAWING 

CHAPTER  I 
INTRODUCTORY 

ART.  PAGE 

101.  Nature  of  drawing 3 

102.  Science  and  art  of  drawing 4 

103.  Magnitude  of  objects 4 

104.  Commercial  application  of  drawing 4 

CHAPTER  II 
OBLIQUE  PROJECTION 

201.  Nature  of  oblique  projection 6 

202.  Oblique  projection  of  lines  parallel  to  the  plane  of  projection 7 

203.  Oblique  projection  considered  as  a  shadow 8 

204.  Oblique  projection  of  lines  perpendicular  to  the  plane  of  projection .  9 

205.  Oblique  projection  of  the  combination  of  parallel  and  perpendicular 

lines  to  the  plane  of  projection 10 

206.  Oblique  projection  of  circles 11 

207.  Oblique  projection  of  inclined  lines  and  angles 12 

208.  Representation  of  visible  and  invisible  lines 13 

209.  Drawings  to  scale 14 

210.  Examples  of  oblique  projection 14 

211.  Distortion  of  oblique  projection 20 

212.  Commercial  application  of  oblique  projection 21 

CHAPTER  III 
ORTHOGRAPHl'c  PROJECTION 

301.  Nature  of  orthographic  projection 25 

302.  Theory  of  orthographic  projection 26 

303.  Revolution  of  the  horizontal  plane 27 

v 


vi  CONTENTS 

ART.  PAGE 

304.  Position  of  the  eye 27 

305.  Relation  of  size  of  object  to  size  of  projection   28 

306.  Location  of  object  with  respect  to  the  planes  of  projection 28 

307.  Location  of  projections  with  respect  to  each  other 29 

308.  Dimensions  on  a  projection 29 

309.  Comparison  between  oblique  and  orthographic  projection 29 

310.  Orthographic  projection  considered  as  a  shadow 30 

311.  Profile  plane 30 

312.  Location  of  profiles 31 

313.  Section  plane 33 

314.  Supplementary  plane 34 

315.  Angles  of  projection 36 

316.  Location  of  observer  in  constructing  projections 36 

317.  Application  of  angles  of  projections  to  drawing 37 

318.  Commercial  application  of  orthographic  projection 38 

CHAPTER  IV 
AXONOMETRIC  PROJECTION 

401.  Nature  of  isometric  projections 45 

402.  Theory  of  isometric  projection 46 

403.  Isometric  projection  and  isometric  drawing 47 

404.  Direction  of  axes 48 

405.  Isometric  projection  of  circles 48 

406.  Isometric  projection  of  inclined  lines  and  angles 49 

407.  Isometric  graduation  of  a  circle 49 

408.  Examples  of  isometric  drawing 51 

409.  Dimetric  projection  and  dime  trie  drawing 54 

410.  Trimetric  projection  and  trimetric  drawing 55 

411.  Axonometric  projection  and  axonometric  drawing 56 

412.  Commercial  application  of  axonometric  projection 56 

413.  Classification  of  projections 57 


PART   II 

GEOMETRICAL    PROBLEMS  IN   ORTHOGRAPHIC* 
PROJECTION 

CHAPTER  V 
REPRESENTATION  OF  LINES  AND  POINTS 

501.  Introductory 61 

502.  Representation  of  the  line 62 

503.  Line  fixed  in  space  by  its  projections 64 

504.  Orthographic  representation  of  a  line 65 

505.  Transfer  of  diagrams  from  orthographic  to  oblique  projection, 66 


CONTENTS  vii 

ART.  PAGE 

506.  Piercing  points  of  lines  on  the  principal  planes 66 

507.  Nomenclature  of  projections 68 

508.  Representation  of  points 68 

509.  Points  lying  in  the  principal  planes 69 

510.  Mechanical  representation  of  the  principal  planes 69 

511.  Lines  parallel  to  the  planes  of  projection 70 

512.  Lines  lying  in  the  planes  of  projection 71 

513.  Lines  perpendicular  to  the  planes  of  projection 72 

514.  Lines  in  all  angles 72 

515.  Lines  with  coincident  projections 74 

516.  Points  in  all  angles 75 

517.  Points  with  coincident  projections 75 

518.  Lines  in  profile  planes 75 

CHAPTER  VI 
REPRESENTATION  OF  PLANES 

601.  Traces  of  planes  parallel  to  the  principal  planes 80 

602.  Traces  of  planes  parallel  to  the  ground  line , 80 

603.  Traces  of  planes  perpendicular  to  one  of  the  principal  planes 82 

604.  Traces  of  planes  perpendicular  to  both  principal  planes 83 

605.  Traces  of  planes  inclined  to  both  principal  planes 83 

606.  Traces  of  planes  intersecting  the  ground  line 84 

607.  Plane  fixed  in  space  by  its  traces 84 

608.  Transfer  of  diagrams  from  orthographic  to  oblique  projection 84 

609.  Traces  of  planes  in  all  angles 86 

610.  Projecting  plane  of  lines 86 

CHAPTER  VII 
ELEMENTARY  CONSIDERATIONS  OF  LINES  AND  PLANES 

701.  Projection  of  lines  parallel  in  space 89 

702.  Projection  of  lines  intersecting  in  space 89 

703.  Projection  of  lines  not  intersecting  in  space 90 

704.  Projection  of  lines  in  oblique  planes 91 

705.  Projection  of  lines  parallel  to  the  principal  planes  and  lying  in  an 

oblique  plane 92 

706.  Projection  of  lines  perpendicular  to  given  planes 95 

707.  Revolution  of  a  point  about  a  line 96 

CHAPTER  VIII 

PROBLEMS  INVOLVING  THE  POINT,  THE  LINE,  AND  THE 

PLANE 

801.  Introductory 99 

802.  Solution  of  problems 99 

803.  Problem  1.  To  draw  a  line  through  a  given  point  parallel  to  a  given 

line.  .                                                                                                  .  100 


viii  CONTENTS 


804.  Problem  2.  To  draw  a  line  intersecting  a  given  line  at  a  giver^  point  ..    100 

805.  Problem  3.    To   find   where    a   given  line  pierces    the    principal 

planes  ................................................    101 

806.  Problem  4.  To  pass  an  oblique  plane  through  a  given  oblique  line.  .    102 

807.  Special  cases  of  the  preceding  problem  .........................   102 

808.  Problem  5.  To  pass  an  oblique  plane  through  a  given  point  .......    103 

809.  Problem  6.  To  find  the  intersection  of  two  planes,  oblique  to  each 

other  and  to  the  principal  planes  ...........................   104 

810.  Special  case  of  the  preceding  problem  ...........................   104 

811.  Problem  7.  To  find  the  corresponding  projection  of  a  given  point 

lying  in  a  given  oblique  plane,  when  one  of  its  projections  is 
given  ...................................................   104 

812.  Special  case  of  the  preceding  problem  ..........................   105 

813.  Problem  8.  To  draw  a  plane  which  contains  a  given  point  and  is 

parallel  to  a  given  plane  ..................................   106 

814.  Problem  9.  To  draw  a  line  perpendicular  to  a  given  plane  through 

a  given  point  ............................................   107 

815.  Special  case  of  the  preceding  problem  ..........................   108 

816.  Problem  10.  To  draw  a  plane  through  a  given  point  perpendicular  to 

a  given  line  .............................................   108 

817.  Problem  11.  To  pass  a  plane  through  three  given  points  not  in  the 

same  straight  line  ........................................   109 

818.  Problem  12.  To  revolve  a  given  point,  not  in  the  principal  planes, 

about  a  line  lying  in  one  of  the  principal  planes  ................   110 

819.  Problem  13.  To  find  the  true  distance  between  two  points  in  space 

as  given  by  their  projections.     First  method.     Case  1  ........   Ill 

820.  Case  2  ...............................  .....................   112 

821.  Problem  13.  To  find  the  true  distance  between  two  points  in  spdce 

as  given  by  their  projections.     Second    method.     Case  1  .  ..    113 

822.  Case  2  .....................................................   113 

823.  Problem  14.  To  find  where  a  given  line  pierces  a  given  plane  ......   114 

824.  Problem  15.  To  find  the  distance  of  a  given  point  from  a  given 

plane  ...................................................   115 

825.  Problem  16.  To  find  the  distance  from  a  given  point  to  a  given  line.  ...    115 

826.  Problem  17.  To   find  the  angle    between   two    given    intersecting 

lines  ...................................................   116 

827.  Problem  18.  To  find  the  angle  between  two  given  planes  .........   117 

828.  Problem  19.  To  find  the  angle  between  a  given  plane  and  one  of 

the  principal  planes  ......................................   118 

829.  Problem  20.  To  draw  a  plane  parallel  to  a  given  plane  at  a  given 

distance  from  it  .........................................   119 

830.  Problem  21.  To  project  a  given  line  on  a  given  plane  ............   120 

831.  Problem  22.  To  find  the  angle  between  a  given  line  and  a  given 

plane  ..................................................   121 

832.  Problem  23.  To  find  the  shortest  distance  between  a  pair  of  skew 

lines  ...................................................   122 

833.  Application  to  other  problems  .................................   125 


CONTENTS  ix 

i 

ART.  PAGE 

834.  Problem  24.  Through  a  given  point,  draw  a  line  of  a  given  length, 

making  given  angles  with  the  planes  of  projection 125 

835.  Problem  25.  Through  a  given  point,  draw  a  plane,  making  given 

angles  writh  the  principal  planes 127 

836.  Problem  26.  Through  a  given  line,  in  a  given  plane,  draw  another 

line,  intersecting  it  at  a  given  angle 129 

837.  Problem  27.  Through  a  given  line,  in  a  given  plane,  pass  another 

plane  making  a  given  angle  with  the  given  plane 130 

838.  Problem  28.  To  construct  the  projections  of  a  circle  lying  in  a  given 

oblique  plane,  of  a  given  diameter,  its  centre  in  the  plane  being 
known 131 


CHAPTER  IX 

* 

CLASSIFICATION  OF  LINES 

901.  Introductory 144 

902.  Straight  line 144 

903.  Singly  curved  line 144 

904.  Representation  of  straight  and  singly  curved  lines 144 

905.  Circle 145 

906.  Ellipse 146 

907.  Parabola 147 

908.  Hyperbola 147 

909.  Cycloid 148 

910.  Epicycloid 149 

911.  Hypocycloid 150 

912.  Spiral 151 

913.  Doubly  curved  line 151 

914.  Representation  of  doubly  curved  lines 151 

915.  Helix 152 

916.  Classification  of  lines 154 

917.  Tangent 154 

918.  Construction  of  a  tangent 154 

919.  To  find  the  point  of  tangency 155 

920.  Direction  of  a  curve 156 

921.  Angle  between  curves 156 

922.  Intersection  of  lines 156 

923.  Order  of  contact  of  tangents 157 

924.  Osculating  circle 158 

925.  Osculating  plane 158 

926.  Point  of  inflexion.     Inflexional  tangent 159 

927.  Normal 159 

928.  Rectification 159 

929.  Involute  and  Evolute 160 

930.  Involute  of  the  circle. .  .  161 


CONTENTS 


CHAPTER  X 
CLASSIFICATION     OF    SURFACES 

ART.  PAGE 

1001.  Introductory 165 

1002.  Plane  surface 165 

1003.  Conical  surface 166 

1004.  Cone 166 

1005.  Representation  of  the  cone 167 

1006.  To  assume  an  element  on  the  surface  of  a  cone 168 

1007.  To  assume  a  point  on  the  surface  of  a  cone 168 

1008.  Cylindrical  surface 169 

1009.  Cylinder 169 

1010V  Representation  of  the  cylinder 170 

1011.  To  assume  an  element  on  the  surface  of  a  cylinder 170 

1012.  To  assume  a  point  on  the  surface  of  a  cylinder 171 

1013.  Convolute  surface 171 

1014.  Oblique  helicoidal  screw  surface 173 

1015.  Right  helicoidal  screw  surface 174 

1016.  Warped  surface 174 

1017.  Tangent  plane 175 

1018.  Normal  plane 175 

1019.  Singly  curved  surface 176 

1020.  Doubly  curved  surface 176 

1021.  Singly  curved  surface  of  revolution 176 

1022.  Doubly  curved  surface  of  revolution 176 

1023.  Revolution  of  a  skew  line 177 

1024.  Meridian  plane  and  meridian  line 177 

1025.  Surfaces  of  revolution  having  a  common  axis 177 

1026.  Representation  of  the  doubly  curved  surface  of  revolution 178 

1027.  To  assume  a  point  on  a  doubly  curved  surface  of  revolution 178 

1028.  Developable  surface 179 

1029.  Ruled  surface 179 

1030.  Asymptotic  surface 179 

1031.  Classification  of  surfaces , 180 


CHAPTER  XI 

INTERSECTIONS  OF  SURFACES  BY  PLANES,  AND  THEIR 
DEVELOPMENT 

1101.  Introductory 184 

1102.  Lines  of  intersection  of  solids  by  planes 185 

1103.  Development  of  surfaces 185 

1104.  Developable  surfaces 185 

1105.  Problem  1.  To  find   the  line  of  intersection  of  the  surfaces  of  a 

right  octagonal  prism  with  a  plane  inclined  to  its  axis 186 


CONTENTS  xi 

j 

ART.  PAGE 

1106.  Problem  2.  To  find  the  developed  surfaces  in  the  preceding  prob- 

lem     187 

1107.  Problem  3.  To  find  the  line  of  intersection  of  the  surface  of  a  right 

circular  cylinder  with  a  plane  inclined  to  its  axis 188 

1108.  Problem  4.  To  find  the  developed  surface  in  the  preceding  problem .  189 

1109.  Application  of  cylindrical  surfaces 189 

1110.  Problem  5.  To  find  the  line  of  intersection  of  the  surfaces  of  a 

right  octagonal  pyramid  with  a  plane  inclined 'to  its  axis.  .  .    190 

1111.  Problem  6.  To  find  the  developed  surf  aces  in  the  preceding  problem  191 

1112.  Problem  7.  To  find  the  line  of  intersection  of  the  surface  of  a  right 

circular  cone  with  a  plane  inclined  to  its  axis 192 

11 13.  Problem  8.  To  find  the  developed  surface  in  the  preceding  problem .  193 

1114.  Application  of  conical  surfaces 194 

1115.  Problem  9.  To  find  the  line  of  intersection  of  a  doubly  curved 

surface  of  revolution  with  a  plane  inclined  to  its  axis 194 

1116.  Problem  10.  To  find  the  line  of  intersection  of  a  bell-surface  with  a 

plane 195 

1117.  Development  by  triangulation 196 

1118.  Problem    11.  To  develop  the  surfaces   of   an   oblique   hexagonal 

pyramid 196 

1119.  Problem  12.  To  develop  the  surface  of  an  oblique  cone 197 

1120.  Problem  13.  To  develop  the  surface  of  an  oblique  cylinder 198 

1121.  Transition  pieces 199 

1122.  Problem  14.  To  develop  the  surface  of  a  transition  piece  connect- 

ing a  circular  opening  with  a  square  opening 200 

1123.  Problem  15.  To  develop  the  surface  of  a  transition  piece  connect- 

ing two  elliptical  openings  whose  major  axes  are  at  right  angles 

to  each  other 201 

1124.  Development  of  doubly  curved  surfaces  by  approximation 202 

1125.  Problem  16.  To  develop  the    surface  of  a  sphere  by  the  gore 

method 203 

1126.  Problem  17.  To  develop  the  surface  of    a  sphere  by  the  zone 

method 204 

1127.  Problem  18.  To  develop  a  doubly  curved  surface  of  revolution  by 

the  gore  method 205 


CHAPTER  XII 

INTERSECTIONS  OF  SURFACES  WITH  EACH  OTHER  AND 
THEIR  DEVELOPMENT 

1201.  Introductory 210 

1202.  Problem  1.  To  find  the  line  of  intersection  of  the  surfaces  of  two 

prisms 211 

1203.  Problem  2.  To  find  the  developments  in  the  preceding  problem. .  -.   211 

1204.  Problem  3.  To  find  the  line  of  intersection  of  two  cylindrical  sur- 

faces of  revolution  whose  axes  intersect  at  a  right  angle  ....    212 


xii  CONTENTS 

ART.  PAGE 

1205.  Problem  4.  To  find  the  developments  in  the  preceding  problem .  .  .  .   213 

1206.  Problem  5.  To  find  the  line  of  intersection  of   two   cylindrical 

surfaces  of  revolution  whose  axes  intersect  at  any  angle.  .  213 

1207.  Problem  6.  To  find  the  developments  in  the  preceding  problem. .  .  214 

1208.  Application  of  intersecting  cylindrical  surfaces  to  pipes 214 

1209.  Problem  7.  To  find  the  line  of  intersection  of  two  cylindrical  sur- 

faces whose  axes  do  not  intersect 215 

1210.  Problem  8.  To  find  the  developments  in  the  preceding  problem. .  .   215 

1211.  Intersection  of  conical  surfaces 216 

1212.  Problem  9.  To  find  the  line  of  intersection  of  the  surfaces  of  two 

cones  whose  bases  may  be  made  to  lie  in  the  same  plane,  and 
whose  altitudes  differ 216 

1213.  Problem  10.  To  find  the  line  of  intersection  of  the  surfaces  of  two 

cones  whose  bases  may  be  made  to  lie  in  the  same  plane,  and 
whose  altitudes  are  equal 218 

1214.  Problem  11.  To  find  the  line  of  intersection  of  the  surfaces  of  two 

cones  whose  bases  lie  in  different  planes 219 

1215.  Types  of  lines  of  intersection  for  surfaces  of  cones 221 

1216.  Problem  12.  To  find  the  line  of  intersection  of  the  surfaces  of  a 

cone  and  a  cylinder  of  revolution  when  their  axes  intersect  at 

a  right  angle 222 

1217.  Problem  13.  To  find  the  line  of  intersection  of  the  surfaces  of  a  cone 

and  a  cylinder  of  revolution  when  their  axes  intersect  at  any 
angle 222 

1218.  Problem  14.  To  find  the  line  of  intersection  of  the  surfaces  of  an 

oblique  cone  and  a  right  cylinder 223 

1219.  Problem  15.  To  find  the  developments  in  the  preceding  problem  .    224 

1220.  Problem  16.  To  find  the  line   of  intersection  of  the  surfaces  of  an 

oblique  cone  and  a  sphere 224 

1221.  Problem  17.  To  find  the  line  of  intersection  of  the  surfaces  of  a 

cylinder  and  a  sphere 225 

1222.  Problem  18.  To  find  the  line  of  intersection  of  two  doubly  curved 

surfaces  of  revolution  whose  axes  intersect 226 

1223.  Commercial  application  of  methods 226 


CONTENTS  xiii 


PART   III 

THE  PRINCIPLES  OF  CONVERGENT  PROJECTING-LINE 

DRAWING 

CHAPTER  XIII 
PERSPECTIVE   PROJECTION 

ART.  PAGE 

1301.  Introductory 235 

1302.  Scenographic  projection 235 

1303.  Linear  perspective 236 

1304.  Visual  rays  and  visual  angle 236 

1305.  Vanishing  point 237 

1306.  Theory  of  perspective  projection 237 

1307.  Aerial  perspective 237 

1308.  Location  of  picture  plane 237 

1309.  Perspective  of  a  line 238 

1310.  Perspectives  of  lines  perpendicular  to  the  horizontal  plane 239 

1311.  Perspectives  of  lines  parallel  to  both  principal  planes 239 

1312.  Perspectives  of  lines  perpendicular  to  the  picture  plane 240 

1313.  Perspectives  of  parallel  lines,  inclined  to  the  picture  plane 240 

1314.  Horizon 241 

1315.  Perspective  of  a  point 242 

1316.  Indefinite  perspective  of  a  line 242 

1317.  Problem  1.  To  find  the  perspective  of  a  cube  by  means  of  the 

piercing  points  of  the  visual  rays  on  the  picture  plane 244 

1318.  Perspectives  of  intersecting  lines 245 

1319.  Perpendicular  and  diagonal 245 

1320.  To  find  the  perspective  of  a  point  by  the  method  of  perpendiculars 

and  diagonals 246 

1321.  To  find  the  perspective  of  a  line  by  the  method  of  perpendiculars 

and  diagonals 248 

1322.  Revolution  of  the  horizontal  plane 249 

1323.  To  find  the  perspective  of  a  point  when  the  horizontal  plane  is 

revolved 249 

1324.  To  find  the  perspective  of  a  line  when  the  horizontal  plane  is 

revolved 250 

1325.  Location  of  diagonal  vanishing  points 251 

1326.  Problem  2.  To  find  the  perspective  of  a  cube  by  the  method  of 

perpendiculars  and  diagonals 251 

1327.  Problem  3.  To  find  the  perspective  of  a  hexagonal  prism 253 

1328.  Problem  4.  To  find  the  perspective  of  a  pyramid  superimposed  on 

a  square  base 254 

1329.  Problem  5.  To  find  the  perspective  of  an  arch 254 

1330.  Problem  6.  To  find  the  perspective  of  a  building 256 

1331.  Commercial  application  of  perspective 258 

1332.  Classification  of  projections .   260 


xiv  CONTENTS 

PART  IV 

PICTORIAL  EFFECTS  OF   ILLUMINATION 
CHAPTER  XIV 

PICTORIAL  EFFECTS  OF  ILLUMONATION  IN  ORTHO- 
GRAPHIC PROJECTION 

ART.  PAGE 

1401.  Introductory 265 

1402.  Line  shading  applied  to  straight  lines 265 

1403.  Line  shading  applied  to  curved  lines 263 

1404.  Line  shading  applied  to  sections 267 

1405.  Line  shading  applied  to  convex  surfaces 267 

1406.  Line  shading  applied  to  concave  surfaces 268 

1407.  Line  shading  applied  to  plane  surfaces 268 

1408.  Physiological  effect  of  light 268 

1409.  Conventional  direction  of  light  rays 269 

1410.  Shade  and  shadow 269 

1411.  Umbra  and  penumbra 269 

1412.  Application  of  the  physical  principles  of  light  to  drawing 270 

1413.  Shadows  of  lines 270 

1414.  Problem  1.  To  find  the  shadow  cast  by  a  cube  which  rests  on  a 

plane 271 

1415.  Problem  2.  To  find  the  shadow  cast  by  a  pyramid,  in  the  principal 

planes 272 

1416.  Problem  3.  To  find  the  shade  and  shadow  cast  by  an  octagonal 

prism  having  a  superimposed  octagonal  cap 273 

1417.  Problem  4.  To  find  the  shade  and  shadow  cast  by  a  superimposed 

circular  cap  on  a  cylinder 275 

1418.  High-light 275 

1419.  Incident,  and  reflected  rays 276 

1420.  Problem  5.  To  find  the  high  light  on  a  sphere 276 

1421.  Multiple  high  lights 277 

1422.  High  lights  on  cylindrical  or  conical  surfaces 277 

1423.  Aerial  effect  of  illumination 277 

1424.  Graduation  of  shade -. 278 

1425.  Shading  rules 278 

1426.  Examples  of  graduated  shades 279 

CHAPTER  XV 

PICTORIAL  EFFECTS  OF  ILLUMINATION  IN  PERSPECTIVE 
PROJECTION 

1501.  Introductory 282 

1502.  Problem  1.  To  draw  the  perspective  of  a  rectangular  prism  and  its 

shadow  on  the  horizontal  plane 282 

1503.  General  method  of  finding  the  perspective  of  a  shadow 284 


CONTENTS  xv 

{ 

ABT.  PAGE 

1504.  Perspectives  of  parallel  rays  of  light 284 

1505.  Perspective  of  the  intersection  of  the  visual  plane  on  the  plane 

receiving  the  shadow 285 

1506.  Application  of  the  general  method  of  finding  the  perspective  of  a 

shadow 285 

1507.  Problem  2.  To  draw  the  perspective  of  an  obelisk  with  its  shade 

and  shadow 286 

1508.  Commercial  application  of  the  pictorial  effects  of  illumination  in 

perspective 289 


PART    I 


PART  I 

PRINCIPLES    OF    PARALLEL    PROJECTING-LINE 

DRAWING 


CHAPTER  I 

INTRODUCTORY 

101.  Nature  of  drawing.  Drawing  has  for  its  purpose  the 
exact  graphic  representation  of  objects  in  -  space.  The  first 
essential  is  to  have  an  idea,  and  then  a  desire  to  express  it.  Ideas 
may  be  expressed  in  words,  in  pictures,  or  in  a  combination  of 
both  words  and  pictures.  If  words  alone  are  sufficient  to  express 
the  idea,  then  language  becomes  the  vehicle  of  its  transmission. 
When  the  idea  relates  to  some  material  object,  however,  a  drawing 
alone,  without  additional  information  may  satisfy  its  accurate 
conveyance.  Further,  some  special  cases  require  for  their  expres- 
sion a  combination  of  both  language  and  drawing. 

Consider,  for  purposes  of  illustration,  a  maple  block,  2  inches 
thick,  4  inches  wide  and  12  inches  long.  It  is  easy  to  conceive  this 
block  of  wood,  and  the  mere  statement,  alone,  specifies  the  object 
more  or  less  completely.  On  the  other  hand,  the  modern  news- 
paper printing  press  can  not  be  completely  described  by  language 
alone.  Anyone  who  has  ever  seen  such  a  press  in  operation, 
would  soon  realize  that  the  intricate  mechanism  could  not  be 
described  in  words,  so  as  to  make  it  intelligible  to  another  without 
the  use  of  a  drawing.  Even  if  a  drawing  is  employed  in  this 
latter  case,  the  desired  idea  may  not  be  adequately  presented, 
since  a  circular  shaft  is  drawn  in  exactly  the  same  way,  whether 
it  be  made  of  wood,  brass,  or  steel.  Appended  notes,  in  such 
cases,  inform  the  constructor  of  the  material  to  use.  From 
the  foregoing,  it  is  evident,  that  drawing  cannot  become  a  uni- 
versal language  in  engineering,  unless  the  appended  descriptions 
and  specifications  have  the  same  meaning  to  all. 


4  PARALLEL  PROJECTING-LINE  DRAWING 

102.  Science  and  art  of  drawing.    Drawing  is  both  a  science 
and   an   art.     The  science   affects  such   matters   as  the   proper 
arrangement  of  views   and   the    manner   of  their  presentation. 
Those  who  are  familiar  with  the  mode  of  representation  used, 
will  obtain  the  idea  the  maker  desired  to  express.     It  is  a  science, 
because  the  facts   can  be  assimilated,  classified,  and  presented 
in  a  more  or  less  logical  order.     In  this  book,  the  science  of 
drawing  will  engage  most  of  the   attention;    only  such  of  the 
artistic  side  is  included  as  adds  to  the  ease  of  the  interpretation 
of  the  drawing. 

The  art  lies  in  the  skilful  application  of  the  scientific  prin- 
ciples involved  to  a  definite  purpose.  It  embraces  such  topics 
as  the  thickness  or  weight  of  lines,  whether  the  outline  alone  is 
to  be  drawn,  or  whether  the  object  is  to  be  colored  and  shaded 
so  as  to  give  it  the  same  appearance  that  it  has  in  nature. 

103.  Magnitude  of  objects.     Objects  visible  to  the  eye  are, 
of  necessity,   solids,  and   therefore   require   the   three   principal 
dimensions  to   indicate   their  magnitude — length,  breadth,    and 
thickness.     If  the  observer  places  himself  in  the  proper  position 
while  viewing  an  object  before  him,  the  object  impresses  itself 
on  him  as  a  whole,  and  a  mental  estimate  is  made  from  the  one 
position  of  the  observer  as  to  its  form  and  magnitude.     Naturally, 
the  first  task  will  be  to  represent  an  object  in  a  single  view,  show- 
ing it  in  three  dimensions,  as  a  solid. 

104.  Commercial   application  of  drawing.     It   must  be 
remembered   that   the    function    of    drawing   is    graphically   to 
present   an  idea  on  a  flat  surface — like   a  sheet  of  paper  for 
instance — so  as  to  take  the  place  of  the  object  in  space.     The 
reader's  imagination  supplies  such  deficiency  as  is  caused  by  the 
absence   of   the   actual   object.     It   is,    therefore,    necessary   to 
study  the  various   underlying  principles  of   drawing,  and,  then, 
apply  them  as  daily  experience  dictates  to  be  the  most  direct 
and  accurate  way  of  their  presentation.     In  any  case,  only  one 
interpretation  of  a  drawing  should  be  possible,  and  if  there  is 
a  possibility  of  ambiguity  arising,  then  a  note  should  be  made 
on  the  drawing  calling  attention  to  the  desired  interpretation. 


INTRODUCTORY 


QUESTIONS  ON  CHAPTER  I 

1.  In  what  ways  may  ideas  be  transmitted  to  others? 

2.  What  topics  are  embraced  in  the  science  of  drawing? 

3.  What  topics  are  included  in  the  art  of  drawing? 

4.  How  many  principal  dimensions  are  required  to  express  the  magni- 

tude of  objects?    What  are  they? 

5.  What  is  the  function  of  drawing? 

6.  Is  the  reader's  imagination  called  upon  when  interpreting  a  drawing? 

Why? 


CHAPTER  II 
OBLIQUE  PROJECTION 

201.  Nature  of  oblique  projection.  Suppose  it  is  desired 
to  draw  a  box,  6"  wide,  12"  long  and  4"  high,  made  of  wood 
\"  thick.  Fig.  1  shows  this  drawn  in  oblique  projection.  The 
method  of  making  the  drawing  will  first  be  shown  and  then 
the  theory  on  which  it  is  based  will  be  developed.  A  rectangle 
abed,  4"X6",  is  laid  out,  the  6"  side  being  horizontal  and  the 
4"  side'  being  vertical.  From  three  corners  of  the  rectangle, 


FIG.  1. 


lines  ae,  bf,  and  eg  are  drawn,  making,  in  this  case,  an  angle  of 
30°  with  the  horizontal.  The  length  12"  is  laid  off  on  an 
inclined  line,  as  eg.  The  extreme  limiting  lines  of  the  box  are 
then  fixed  by  the  addition  of  two  lines  ef  (horizontal)  and  fg 
(vertical).  The  thickness  of  the  wood  is  represented,  and  the 
dimensions  showing  that  it  is  \"  thick  indicate  the  direction 
in  which  they  are  laid  off.  The  reason  for  the  presence 
of  such  other  additional  lines,  is  that  they  show  the  actual 
construction. 

The  sloping  lines  in  Fig.  1  could  be  drawn  at  any  angle  other 

6 


OBLIQUE  PROJECTION 


than  30°.    In  Fig.  2,  the  same  box  is  drawn  with  a  60°  inclina- 

tion.    It  will  be  seen,  in  this  latter  case,  that  the  inside  bottom 

of     the     box     is     also     shown 

prominently.      It    is    customary 

in  the  application  of  this  type 

of  drawing,   to  use   either    30°, 

45°  or  60°  for  the  slope,  as  these 

lines  can  be  easily  drawn  with 

the    standard   triangles   used  in  > 

the  drafting  room. 

202.  Oblique  projection  of 
lines  parallel  to  the  plane  of 
projection.  In  developing  the 
theory,  let  XX  and  YY,  Fig.  3,  be 
two  planes  at  right  angles  to  each 
other.  Also,  let  ABCD  be  a  thin 
rectangular  plate,  the  plane  of  FIG.  2. 

which  is  parallel  to  the  plane  XX. 

Suppose  the  eye  is  looking  in  the  direction  Aa,  inclined*  to  the 
plane  XX.     Where  this  line  of  sight  from  the  point  A  on  the 


FIG.  3. 

object  appears  to  pierce  or  impinge  on  the  plane  XX,  locate  the 
point  a.    From  the  point  B,  assume  that  the  eye  is  again  directed 

*  The  ray  must  not  be  perpendicular,  as  this  makes  it  an  orthographic 
projection.  The  ray  cannot  be  parallel  to  the  plane  of  projection,  because 
it  will  never  meet  it,  and,  hence,  cannot  result  in  a  projection. 


8  PARALLEL  PROJECTING-LINE  DRAWING 

toward  the  plane  XX,  in  a  line  that  is  parallel  to  Aa;  this 
second  piercing  point  for  the  point  B  in  space  will  appear 
at  b.  Similarly,  from  the  points  C  and  D  on  the  object,  the 
piercing  points  ron  [the  plane  will  be  c  and  d,  Cc  and  Dd  being 
parallel  to  Aa. 

On  the  plane  XX,  join  the  points  abed.  To  an  observer, 
the  figure  abed  will  give  the  same  mental  impression  as  will 
the  object  ABCD.  In  other  words,  abed  is  a  drawing  of  the 
thin  plate  ABCD.  ABCD  is  the  object  in  space;  abed  is  the 
corresponding  oblique  projection  of  ABCD.  The  plane  XX  is 
the  plane  of  projection;  Aa,  Bb,  Cc,  and  Dd,  are  the  projecting 
lines,  making  any  angle  with  the  plane  of  projection  other  than 
at  right  angles  or  parallel  thereto.  The  plane  YY  serves  the 
purpose  of  throwing  the  plane  XX  into  stronger  relief  and  has 
nothing  to  do  with  the  projection. 

It  will  be  observed  that  the  figure  whose  corners  are  the 
points  ABCDabcd  is  an  oblique  rectangular  prism,  the  opposite 
faces  of  which  are  parallel  because  the  edges  have  been  made 
parallel  by  construction.  From  the  geometry,  all  parallel  plane 
sections  of  the  prism  are  equal,  hence  abed  is  equal  to  ABCD, 
because  the  plane  of  the  object  ABCD  was  originally  assumed 
parallel  to  XX,  and  the  projection  abed  lies  in  the  plane  XX. 
As  a  corollary,  the  distance  of  the  object  from  the  plane  of  pro- 
jection does  not  influence  the  size  of  the  projection,  so  long  as 
the  plane  of  the  object  is  continually  parallel  to  the  plane  of 
projection. 

Indeed,  any  line,  whether  straight  or  curved,  when  parallel 
to  the  plane  of  projection  has  its  projection  equal  to  the  line 
itself.  This  is  so  because  the  curved  line  may  be  considered 
as  made  up  of  an  infinite  number  of  very  short  straight  lines. 

203.  Oblique  projection  considered  as  a  shadow.  Another 
way  of  looking  at  the  projection  shown  in  Fig.  3  is  to  assume 
that  light  comes  in  parallel  lines,  oblique  to  the  plane  of  pro- 
jection. If  the  object  is  interposed  in  these  parallel  rays,  then 
abed  is  the  shadow  of  A^CD  in  space,  and  thus  presents  an 
entirely  different  standpoint  from  which  to  consider  the  nature 
of  a  projection.  Both  give  identical  results,  and  the  latter  is 
here  introduced  merely  to  reenforce  the  understanding  of  the 
nature  of  the  operation. 


OBLIQUE  PROJECTION 


9 


204.  Oblique  projection  of  lines  perpendicular  to  the 
plane  of  projection.  Let  XX,  Fig.  4,  be  a  transparent  plane 
surface,  seen  edgewise,  and  ab,  an  arrow  perpendicular  to  XX, 
the  end  a  of  the  arrow  lying  in  the  plane.  Suppose  the  eye  is 
located  at  r  so  that  the  ray  of  light  rb  makes  an  angle  of  45° 
with  the  plane  XX.  If  all  rays  of  light  from  points  on  ab  are 


x 
FIG.  4. 


FIG.  5. 


parallel  to  rb,  they  will  pierce  the  plane  XX  in  a  series  of  points, 
and  ac,  then,  will  become  the  projection  of  ab  on  the  plane  XX. 
To  an  observer  standing  in  the  proper  position,  looking  along 
lines  parallel  to  rb,  ac  will  give  the  same  mental  impression  as 
the  actual  arrow  ab  in  space;  and,  therefore,  ac  is  the  projection 
of  ab  on  the  plane  XX.  The  extremities  of  the  line  ab  are  hence 
projected  as  two  distinct  points  a  and  c.  Any  intermediate 
point,  as  d,  will  be  projected  as  e  on  the  projection  ac. 

From  the  geometry,  it  may  be  noticed  that  ac  is  equal  to  ab, 
because  cab  is  a  right  angled  triangle,  and  the  angle  acb  equals 
the  angle  cba,  due  to  the  adoption  of  the  45°  ray.  Also,  any 
limited  portion  of  a  line,  perpendicular  to  the  plane  of  projection, 
is  projected  as  a  line  equal  to  it  in  length.  The  triangle  cab  may 
be  rotated  about  ab  as  an  axis,  so  that  c  describes  a  circle  in 
the  plane  XX  and  thus  ac  will  always  remain  equal  to  ab.  This 
means  that  the  rotation  merely  corresponds  to  a  new  position 
of  the  eye,  the  inclination  of  the  ray  always  remaining  45°  with 
the  plane  XX. 


10 


PARALLEL  PROJECTING-LINE  DRAWING 


The  foregoing  method  of  representing  45°  rays  is  again  shown 
as  an  oblique  projection  in  Fig.  5.  Two  positions  of  tiie  ray 
are  indicated  as  rb  and  sb;  the  corresponding  projections  are 
ac  and  ad.  Hence,  in  constructing  oblique  projections,  the 
lines  that  are  parallel  to  the  plane  of  projection  are  drawn  with 
their  true  relation  to  each  other.  The  lines  that  are  perpendicular 
to  the  plane  of  projection  are  drawn  as  an  inclined  line  of  a  length 
'equal  to  tke  line  itself  and  making  any  angle  with  the  horizontal, 


FIG.  6. 


at  pleasure.  Here,  again,  the  plane  YY  is  added.  The  line  of 
intersection  of  XX  and  YY  is  perpendicular  to  the  plane  of 
the  paper,  and  is  shown  as  a  sloping  line,  because  the  two  planes 
themselves  are  pictured  in  oblique  projection.* 

205.  Oblique  projection  of  the  combination  of  parallel 
and  perpendicular  lines  to  the  plane  of  projection.  Fig. 
6  shows  a  box  and  its  projection,  pictorially  indicating  all  the 
mental  steps  required  in  the  construction  of  an  oblique  pro- 
jection. The  object  (a  box)  A  is  shown  as  an  oblique  projection; 

*  Compare  this  with  Figs.  1  and  2.  The  front  face  of  the  box  is  shown 
as  it  actually  appears,  because  it  is  parallel  to  the  plane  of  projection  (or 
paper).  The  length  of  the  box  is  perpendicular  to  the  plane  of  the  paper 
and  is  projected  as  a  sloping  line. 


OBLIQUE   PROJECTION 


11 


its  projection  B  on  the  plane  of  projection  appears  very  much 
distorted.  This  distortion  of  the  projection  is  due  to  its  being 
an  oblique  projection,  initially,  which  is  then  again  shown  in 
oblique  projection.  From  what  precedes,  the  reader  should  find 
no  difficulty  in  tracing  out  the  construction.  Attention  may 
again  be  called  to  the  fact  that  the  extremities  of  the  lines  per- 
pendicular to  the  plane  of  projection  are  projected  as  two  distinct 
points. 

206.  Oblique  projection  of  circles.*  When  the  plane  of  a 
circle  is  parallel  to  the  plane  of  projection,  it  is  drawn  with  a 
compass  in  the  ordinary  way,  because  the  projection  is  equal 
to  the  circle  itself  (202).  When  the  plane  of  the  circle  is  per- 
pendicular to  the  plane  of 
projection,  however,  it  is 
shown  as  an  ellipse.  Both 
cases  will  be  illustrated  by 
Fig.  7,  which  shows  a  cube 
in  oblique  projection.  In 
the  face  abed,  the  circle  is 
shown  as  such,  because  the 
plane  of  the  circle  is  par- 
allel to  the  plane  of  pro- 
jection, which,  in  this  case, 
is  the  plane  of  the  paper.  It 
will  be  noticed  that  the  circle  FlG-  7- 

in  abed  is  tangent  at  points 

midway  between  the  extremities  of  the  lines.  If  similar  points 
of  tangency  are  laid  off  in  the  faces  aefb  and  fbcg  and  a  smooth 
curve  be  drawn  through  these  points,  the  result  will  be  an  ellipse; 
this  ellipse  is,  therefore,  the  oblique  projection  of  a  circle,  whose 
plane  is  perpendicular  to  the  plane  of  projection.  The  additional 
lines  in  Fig.  7  show  how  four  additional  points  may  be  located 
on  the  required  ellipse. 

It  may  be  shown  that  a  circle  is  projected  as  an  ellipse  in  all 
cases  except  when  its  plane  is  parallel  to  the  plane  of  projection, 
or,  when  its  plane  is  chosen  parallel  to  the  projecting  lines.  In 
the  latter  case,  it  is  a  line  of  a  length  equal  to  the  diameter  of  the 

*When  projecting  circles  in  perpendicular  planes,  the  30°  slope  offers 
an  advantage  because  the  ellipse  is  easily  approximated.  See  Art.  405. 


PARALLEL  PROJECTING-LINE   DRAWING 


circle.  The  reason  for  this  will  become  evident  later  in  the- 
subject.  (It  is  of  insufficient  import  at  present  to  dwell  on  it 
at  length.) 

207.  Oblique  projection  of  inclined   lines  and  angles. 

At  times,  lines  must  be  drawn  that  are  neither  parallel  nor 
perpendicular  to  the  plane  of  projection.  A  reference  to  Fig. 
8  will  show  how  this  is  done.  It  is  desired  to  locate  a  hole  in 
a  cube  whose  edge  measures  12".  The  hole  is  to  be  placed  in 
the  side  bfgc,  8"  back  from  the  point  c  and  then  4"  up  to  the 
point  h.  To  bring  this  about,  lay  off  ck  =  8"  and  kh  (vertically) 

=  4"  and,  then  h  is  the 
12"  -  required  point.  Also,  hkc 

is  the  oblique  projection 
of  a  right  angled  tri- 
angle, whose  plane  is  per- 
pendicular to  the  plane 
of  projection.  Suppose, 
further,  it  is  desired  to 
locate  the  point  m  on 
the  face  abfe,  1"  to  the 
right  of  the  point  a  and 
5"  back.  The  dimensions 
show  how  this  is  done. 
Again,  man  is  the  ob- 
lique projection  of  aright 
angled  triangle,  whose 

legs  are  5"  and  1".  This  method  of  laying  off  points  is  virtually 
a  method  of  offsets.*  The  point  m  is  offset  a  distance  of  5" 
from  ab;  likewise  h  is  offset  4"  from  eg.  If  it  be  required  to  lay 
off  the  diagonal  of  a  cube,  it  is  accomplished  by  making  three 
offsets  from  a  given  point.  For  instance,  consider  the  diagonal 
ce.  If  c  is  the  starting  point,  draw  eg  perpendicular  to  thft 
plane  (shown  as  an  inclined  line),  then  fg,  vertically  upward, 
and  finally  fe,  horizontally  to  the  left;  therefore,  ce  is  the  diagonal 
of  the  cube,  if  eg  =  gf  =  fe.f 

*  This  method  is  of  importance  in  that  branch  of  mathematics  known 
as  Vector  Analysis.  Vectors  are  best  drawn  in  space  by  means  of  oblique 
projection. 

t  If  two  given  lines  are  parallel  in  space,  their  oblique  projections  are 
parallel  under  any  conditions.  The  projecting  lines  from  the  extremities  of 


OBLIQUE   PROJECTION 


13 


A  word  may  be  said  in  reference  to  round  holes  appearing 
in  the  oblique  faces  of  a  cube.  As  has  been  shown,  circles  are 
here  represented  as  ellipses  (206),  but  if  it  were  desired  to  cut 
an  elliptical  hole  at  either  h  or  m,  then  their  projections  would 
not  give  a  clear  idea  of  the  fact.  Such  cases,  when  they  occur, 
must  be  covered  by  a  note  to  that  effect;  an  arrow  from  the 
note  pointing  to  the  hole  would  then  indicate,  unmistakably, 
that  the  hole  is  to  be  drilled  (for  a  round  hole),  otherwise  its 
shape  should  be  called  for  in  any  way  that  is  definite.  It  seems, 
therefore,  that  oblique  projection  cannot  fulfill  the  needs  of 
commercial  drawing  in  every  respect;  and,  indeed,  this  is  true. 
Other  methods  also  have  certain  advantages  and  will  be  treated 
subsequently  4 

208.  Representation  of  visible  and  invisible  lines.     While 
viewing  an  object,  the    observer  finds  that  some  lines  on  the 
object  are  visible.     These  lines  are  drawn  in  full  on  the  projection. 
There  are,  however,  other  lines,  invisible  from  the  point  of  view 
chosen  and  these,  when  added,  are  shown  dotted.     Fig.  9  shows 
all  the  visible  and    invisible 
lines    on    a    hollow    circular 
cylinder.       Dimensions     are 
appended    and    the  cylinder 
shaded  so  that  no   question 
should  arise  as  to  its  identity. 
It  can  be  observed  that  the 
drawing  is  clear  in  so  far  as 
it  shows  that  the  hole  goes 
entirely  through    the    cylin- 
der.    Were  the  dotted  lines  FIG.  9. 
omitted,  one  could  not   tell 

whether  the  hole  went  entirely  through,  or  only  part  way 
through.  Hence,  dotted  lines  may  add  to  the  clearness  of  a 
drawing;  in  such  cases  they  should  be  added.  At  times,  how- 
ever, their  addition  may  lead  to  confusion;  and,  then,  only  the 

the  given  lines  determine  planes  that  cut  the  plane  of  projection  in  lines 
which  are  the  projections  of  the  given  lines.  The  projecting  planes  from 
the  given  lines  are  parallel,  and,  hence,  their  projections  are  parallel,  since 
it  is  the  case  of  two  parallel  planes  cut  by  a  third  plane. 

t  The  student  will  obtain  many  suggestions  by  copying  such  simple 
illustrations  as  Figs.  1,  2,  7,  8  and  9. 


14  PARALLEL  PROJECTING-LINE  DRAWING 

more  important  dotted  lines  added,  and  such  others,  considered 
unnecessary,  should  be  omitted.  Practice  varies  in  this  latter 
respect  and  the  judgment  of  the  draftsman  comes  into  play 
at  this  point;  ability  to  interpret  the  drawing  rapidly  and 
accurately  is  the  point  at  issue. 

209.  Drawings  to  scale.     Objects  of  considerable  size  cannot 
be   conveniently  represented   in    their  full   size.     The  shape  is 
maintained,  however,  by  reducing  the  length  of  each  a  definite 
proportion  of  its  original  length,  or,  in  other  words,  by  drawing 
to  scale.     Thus,  if  the  drawing  is  one-half  the  size  of  the  object, 
the  scale  is  6"  =  1  ft.  and  is  so  indicated  on  the  drawing  by  a  note 
to  that  effect.     The  scales  in  common  use  are  12"  =  1  ft.  or  full 
size;   6"  =  1  ft.  or  half-size;    3"  =  1  ft.  or  quarter  size;    1J";    1" 

f";  4";  f";  i";  A";  i";  A";    A";  etc.  =  i  ft.    The  smaller 

sizes  are  used  for  very  large  work  and  vice  versa.  In  railway 
work,  scales  like  100' =  1  inch,  or  10000'  =  !  inch  are  common. 
In  watch  mechanism,  scales  like  48"  =  I  ft.  or  "  four  times  actual 
size "  or  even  larger  are  used,  since,  otherwise,  the  drawings 
would  be  too  small  for  the  efficient  use  of  the  workman.  Irre- 
spective of  the  scale  used,  the  actual  dimensions  are  put  on  the 
drawing  and  the  scale  is  indicated  on  the  drawing  by  a  note 
to  that  effect. 

If  some  dimensions  are  not  laid  out  to  the  scale  adopted, 
the  drawing  may  create  a  wrong  impression  on  the  reader  and 
this  should  be  avoided  if  possible.  When  changes  in  dimen- 
sions occur  after  the  completion  of  a  drawing  and  it  is  im- 
practicable to  make  the  change,  the  dimension  may  be 
underlined  and  marked  conveniently  near  it  N  T  S,  meaning  "  not 
to  scale." 

210.  Examples   of    oblique    projection.     Fig.  10  shows  a 
square  block  with  a  hole  in  its  centre.    The  dimension  lines  indicate 
the  size  and  the  method  of  making  the  drawing  when  the  planes 
of  the  circles  are  chosen  parallel  to  the  plane  of  the  paper  (plane 
of  projection).     The  circle  in  the  visible  face  is  drawn  with  a 
compass  to  the  desired  scale.     The  circle  in  the  invisible  face 
(invisible  from  the  point  of  view  chosen)  is  drawn  to  the  same 
radius,  but,  its  centre  is  laid  off  on  an  inclined  line,  a  distance 
back  of  the  visible  circle,  equal  to  the  thickness  of  the  block. 
The  circle  in  the  front  face  is  evidently  not  in  the  same  plane 


OBLIQUE  PROJECTION 


15 


as  that  in  the  distant  face.  The  line  joining  their  centres  is 
thus  perpendicular  to  the  plane  of  projection,  and  is,  hence, 
laid  off  as  an  inclined  line. 


FIG.  10. 


FIG. 11. 

» 

When  the  plane  of  the  circles  is  made  perpendicular  to  the 
plane  of  projection,  the  circles  are  projected  as  ellipses.  Fig. 
11  shows  how  the  block  of  Fig.  10  is  drawn  when  such  is  the 


16 


PARALLEL  PROJECTING-LINE  DRAWING 


case.  It  is  to  be  observed,  that  the  bounding  square  is  to  be 
drawn  first  and  then  the  ellipse  (projection  of  the  circle)  is  inscribed. 
When  the  circles  in  both  faces  are  to  be  shown,  the  bounding 


FIG.  13. 


square  must  be  replaced  by  a  bounding  rectangular  prism.  This 
rectangular  prism  is  easily  laid  out  and  the  ellipses  are  inserted 
in  the  proper  faces.  The  method  of  using  bounding  figures  of 


OBLIQUE  PROJECTION 


17 


simple  shape  is  of  considerable  importance  when  applying  the 
foregoing  principles  to  oblique  projection. 


FIG.  14. 


FIG.  15. 


Fig.  12    is    another  illustration  of  an  object,  differing  from 
Figs.  10  and  11  in  so  far  as  the  hole  does  not  go  entirely  through 


18 


PARALLEL  PROJECTING-LINE  DRAWING 


from  face  to  face.  The  centres  for  the  different  circles  are  found 
on  the  axis.  The  distances  between  the  centres,  measured  on 
the  inclined  line,  is  equal  to  the  distances  between  the  planes 
of  the  corresponding  circles.  Since  the  circles  are  drawn  as  such, 
the  planes  must,  therefore,  be  parallel  to  the  plane  of  projection. 
The  axis  of  the  hole  is  perpendicular  to  the  plaite  of  projection, 
and,  hence,  is  projected  as  an  inclined  line. 

The  object  in  Fig.  12  is  also  shown  with  the  planes  of  the 
circles  perpendicular  to  the  plane  of  projection  in  Fig.  13.  Every 
step  of  the  construction  is  indicated  in  the  figure  and  the  series 
of  bounding  prisms  about  the  cylinders  is  also  shown. 


FIG.  16. 

A  somewhat  different  example,  showing  the  necessity  of 
bounding  figures,  is  given  in  Fig.  14.  The  bounding  figure, 
shown  in  Fig.  15,  is  laid  out  as  given  and  it  becomes  a  simple 
matter  to  insert  the  object  subsequently.  It  should  be  noted 
how  the  rectangular  projection  becomes  tangent  to  the  cylinder 
and  that  the  only  way  to  be  certain  of  the  accuracy  of  the 
drawing,  is  to  use  these  bounding  figures  and  make  mental 
record  of  the  relative  location  of  the  lines  that  make  up  the 
drawing. 

If  objects  are  to  be  drawn  whose  lines  are  inclined  to  each 
other,  the  principles  so  far  developed  offer  simple  methods  for 
their  presentation.  Fig.  16  shows  a  tetrahedron  with  a  bounding 


OBLIQUE   PROJECTION 


19 


rectangular  prism.  The  apex  is  located  on  the  top  face  and 
its  position  is  determined  from  the  geometric  principles  imposed. 
Solids,  as  represented  in  the  text-books  on  geometry,  are  drawn 
•in  this  way.  Some  confusion  may  be  avoided  by  observing 
that  angles  are  only  preserved  in  their  true  relation  in  the  planes 
parallel  to  the  plane  of  projection  (207). 

The  concluding  example  of  this  series  is  given  in  Fig.  17. 
It  is  known  as  a  bell-crank  and  has  circles  shown  in  two  planes 
at  right  angles  to  each  other,  The  example  furnishes  the  clue 


FIG.  17. 


to  constructing  any  object,  however  complicated  it  may  be. 
Base  lines  ab  and  be  are  first  laid  out  to  the  required  dimensions 
and  to  the  desired  scale.  In  this  example,  the  base  lines  are 
chosen  parallel  to  the  plane  of  projection,  and  hence  are  projected 
as  a  right  angle,  true  to  dimensions.  The  thickness  of  the  two 
lower  cylinders  is  laid  off  as  an  inclined  line  from  each  side  of  the 
base  line  and  the  circles  are  then  drawn.  The  upper  circles 
(shown  as  ellipses)  may  need  some  mention.  A  bounding  rec- 
tangular prism  is  first  drawn,  half  of  which  is  laid  off  on  each 
side  of  the  base  line  (true  in  this  case  but  it  may  vary  in  others). 


20 


PARALLEL  PROJECTING-LINE  DRAWING 


The  circles  and  inclined  lines  are  filled  in  after  the  guiding  details 
are  correctly  located.  This  drawing  may  present  some  difficulty 
at  first,  but  a  trial  at  its  reproduction  will  reveal  no  new  prin- 
ciples, only  an  extreme  application. 

On  completion  of  the  drawing,  the  bounding  figure  may  be 
removed  if  its  usefulness  is  at  an  end.  When  inclined  lines 
appear  frequently  on  the  drawing,  the  bounding  figures  can  be 
made  to  serve  as  dimension  lines,  and  so  help  in  the  interpre- 
tation. The  draftsman  must  determine  what  is  best  in  each 
case,  remembering,  always,  that  the  drawing  must  be  clear 
not  only  to  himself,  but  to  others  who  may  have  occasion  to 
read  it. 

211.  Distortion  of  oblique  projection.  A  view  of  a  com- 
pleted machine  suffers  considerable  distortion  when  drawn  in 


\x 
FIG.  18. 


FIG.  19. 


oblique  projection  because  the  eye  cannot  be  placed  in  any  one  posi- 
tion, whereby  it  can  view  the  drawing  in  the  manner  the  projec- 
tion was  made.  *  To  overcome  this  difficulty  to  some  extent  and 
to  avoid  bringing  the  distortion  forcibly  to  the  attention  of  the 
observer,  the  projecting  lines  can  be  so  chosen  that  the  per- 
pendicular to  the  plane  of  projection  is  projected  as  a  shorter 
line  than  the  perpendicular  itself.  Fig.  18  shows  this  in  con- 
struction. XX  is  a  vertical  transparent  plate,  similar  to  that 
shown  in  Fig.  4.  The  ray  rb  makes  an  angle  with  XX  greater 
than  45°,  and,  by  inspection,  it  is  seen  that  the  projection  of 
ab  on  XX  is  ac,  which  is  shorter  than  ab,  the  perpendicular. 

The  application  of  the  foregoing  reduces  simply  to  this:   All 
lines  and  curves  parallel  to  the  plane  of  projection  are  shown 
*  This  condition  is  satisfied  in  Perspective  Projections. 


OBLIQUE   PROJECTION  21 

exactly  the  same  as  in  oblique  projection  with  45°  ray  inclination. 
The  lines  that  are  perpendicular  to  the  plane  of  projection  are 
reduced  to  i,  |,  J,  etc.  of  their  original  length  and  reduced  or 
increased  to  the  scale  adopted  in  making  the  drawing.  This 
mode  of  representation*  is  suitable  for  making  catalogue  cuts 
and  the  like.  It  gives  a  sense  of  depth  without  very  noticeable 
distortion,  due  to  two  causes:  the  impossible  location  of  the 
eye  while  viewing  the  drawing,  and,  the  knowledge  of  the  apparent 
decrease  in  size  of  objects  as  they  recede  from  the  eye.  A  single 
illustration  is  shown  in  Fig.  19. 

212.  Commercial    application    of    oblique     projection. 

Oblique  projection  is  useful  in  so  far  as  it  presents  the  three 
dimensions  in  a  single  view.  When  curves  are  a  part  of  the 
outline  of  the  object,  it  is  desirable  to  make  the  plane  of  the 
curve  parallel  to  the  plane  of  projection,  thereby  making  the 
projection  equal  to  the  actual  curve  and  also  economizing  time 
in  making  the  drawing.  Sometimes  it  is  not  possible  to  carry 
this  out  completely.  Fig.  17,  already  quoted,  shows  an  example 
of  this  kind.  It  is  quite  natural  to  make  the  drawing  as  shown, 
because  the  planes  of  most  of  the  circles  are  parallel  to  the  plane 
of  projection,  leaving,  thereby,  only  one  end  of  the  bell-crank 
to  be  projected  with  ellipses. 

Oblique  projections,  in  general,  are  perhaps  the  simplest 
types  of  drawings  that  can  be  made,  if  the  objects  are  of  com- 
paratively simple  shape.  They  carry  with  them  the  further 
advantage  that  even  the  uninitiated  are  able  to  read  them,  when 
the  objects  are  not  unusually  intricate.  The  making  of  oblique 
projections  is  simple,  but,  at  the  same  time,  they  call  on  the 
imagination  to  some  extent  for  their  interpretation.  This  is 
largely  due  to  the  fact  that  the  eye  changes  its  position  for  each 
point  projected,  and  that  no  one  position  of  the  eye  will  properly 
place  the  observer  with  respect  to  the  object. 

The  application  of  oblique  projection  to  the  making  of 
drawings  for  solid  geometry  is  already  known  to  the  student 
and  the  resulting  clarity  has  been  noticed.  Other  types  of  pro- 
jections have  certain  advantages  which  will  be  considered  in  due 
order. 

*  This  type  of  projection  has  been  called  Pseudo  Perspective  by  Dr. 
MacCord  in  his  Descriptive  Geometry. 


PARALLEL  PROJECTING-LINE   DRAWING 

The  convenience  of  oblique  projection  to  the  laying  out  of 
piping  diagrams  is  worthy  of  mention.  Steam  and  water  pipes, 
plumbing,  etc.,  when  laid  out  this  way,  result  in  an  exceedingly 
readable  drawing. 

QUESTIONS  ON  CHAPTER  II 

1.  What  is  an  oblique  projection? 

2.  What  is  a. plane  of  projection? 

3.  What  is  a  projecting  line? 

4.  Prove  that  when  a  rectangle  is  parallel  to  the  plane  of  projection  the 

projection  of  the  rectangle  is  equal  to  the  rectangle  itself. 

5.  Does  the  distance  of  the  object  from  the  plane  have  any  influence  on 

the  size  of  the  projection?    Why? 

6.  Prove  that  any  line,  whether  straight  or  curved,  is  projected  in  its 

true  form  when  it  is  parallel  to  the  plane  of  projection. 

7.  Show  under  what  conditions  a  projection  may  be  considered  as  a 

shadow. 

8.  Prove  that  when  a  line  is  perpendicular  to  the  plane  of  projection, 

it  is  projected  as  a  line  of  equal  length,  when  the  projecting  rays 
make  an  angle  of  45°  with  the  plane  of  projection.  Use  a  diagram. 

9.  Prove  that  any  limited  portion  of  a  line  is  projected  as  a  line  of 

equal  length,  when  the  line  is  perpendicular  to  the  plane  of  projec- 
tion and  the  projecting  lines  make  an  angle  of  45°  with  the  plane 
of  projection. 

10.  Show  how  a  perpendicular  may  be  projected  as  a  longer  or  a  shorter 

line,  if  the  angle  of  the  projecting  lines  differs  from  45°. 

11.  Why  can  not  the  projecting  lines  be  selected  parallel  to  the  plane  of 

projection? 

12.  Show  that  when  a  line  is  parallel  to  the  plane  of  projection,  it  is 

projected  as  a. line  of  equal  length,  irrespective  of  the  angle  of  the 
projecting  lines,  provided  the  projecting  lines  are  inclined  to  the 
plane  of  projection. 

13.  Why  may  the  slope  of  the  projection  of  a  line  perpendicular  to  the 

plane  of  projection  be  drawn  at  any  angle? 

14.  Prove  that  when  two  lines  are  parallel  to  each  other  and  also  to  the 

plane  of  projection,  their  projections  are  parallel. 

15.  Prove  that  when  two  lines  are  perpendicular  to  the  plane  of  pro- 

jection, their  projections  are  parallel  to  each  other. 

16.  Draw  two  rectangular  planes  at  right  angles  to  each  other  so  that  the 

edges  of  the  planes  are  parallel  or  perpendicular  to  the  plane  of 
the  paper  (or  projection). 

17.  Draw  a  cube  in  oblique  projection  and  show  which  lines  are  assumed 

parallel  to  the  plane  of  projection  and  which  lines  are  perpendicular 
to  the  plane  of  projection. 

18.  Draw  a  cube  in  oblique  projection  and  show  how  the    circles    are 

inserted  in  each  of  the  visible  faces. 


OBLIQUE  PROJECTION 


23 


19.  Show  how  angles  are  laid  off  on  the  face  of  a  cube  in  oblique  pro- 

jection. 

20.  Under  what  conditions  is  the  angular  relation  between  lines  pre- 

served? 

21.  Draw  a  line  that  is  neither  parallel  nor  perpendicular  to  the  plane 

of   projection.      (Use  the   cube,   in    projection,   as    a   bounding 
figure.) 

22.  Prove  that  any  two  lines  in  space  are  projected  as  parallels  when 

they  themselves  are  parallel. 

23.  Under   what    conditions  will  the  oblique  projection  of  a  line  be  a 

point? 

24.  How  are  visible  and  invisible  lines  represented  on  a  drawing? 

25.  What  is  meant  by  drawing  to  scale? 

26.  Why  is  it  desirable  to  have  all  parts  of  the  same  object  drawn  to 

true  scale? 


v. 


FIG.  2A. 


FIG.  2B. 


27.  What  considerations  govern  the  choice  of  the  scale  to  be  used  on  a 

drawing? 

28.  Show  how  the  distortion  of  an  oblique  projection  may  be  reduced  by 

changing  the  angle  of  the  projecting  lines. 

29.  Why  is  it  impossible  to  locate  the  eye  in  one  position  and  view  the 

projection  in  the  manner  in  which  it  was  made? 

30.  Draw  a  rectangular  box  with  a  hinged  cover,  in  oblique  projection, 

and  show  the  cover  partly  raised. 

31.  Draw  a  lever,  having  a  round  hole  on  one  end  so  as  to  fit  over  a 

shaft.     Have  the  plane  of  the  circles  parallel  to  the  plane  of 
projection. 

32.  Draw  an  oblong  block,    2"x3"x6"  long,   in  oblique  projection, 

having  a  1"  hole  in  its  centre,  4"  deep. 

33.  Draw  a  cylindrical  shaft,  6"  in  diameter  and  18"  long,  in  oblique 

projection,   having  a  rectangular  hole,   2"X3"x5"   deep,   from 
each  end.    Lay  out  to  a  scale  of  3"  =  1  ft.  and  affix  all  dimensions. 


24  PARALLEL  PROJECTING-LINE  DRAWING 

34.  Draw  a  triangular  prism,  in  oblique  projection,  showing  how  the 

bounding  figure  is  used. 

35.  Draw  a  hexagonal  prism,  in  oblique  projection,  and  show  all  the 

invisible  (dotted)  lines  on  it. 

36.  Draw  a  hexagonal  pyramid  in  oblique  projection. 

37.  Make  a  material  list  for  the  box  shown  in  Fig.  2A. 

38.  Draw    the    circular  cylinder,  4"  diameter  and  6"  long,  shown  in 

Fig.  2B.  On  the  surface  of  this  cylinder,  two  semi-circular  grooves 
are  cut,  as  shown  by  the  dimensions.  Make  two  drawings  in 
oblique  projection,  one  showing  the  plane  of  the  circles  parallel 
to  the  plane  of  projection  and  the  other,  with  the  plane  of  the 
circles  perpendicular  to  the  plane  of  projection. 

NOTE. — For  additional  drawing  exercises  see  examples  in  Chapters 
III  and  IV. 


CHAPTER  III 
ORTHOGRAPHIC   PROJECTION 

301.  Nature  of  orthographic  projection.  Take,  for 
example,  a  box  6"X12"X4"  high,  made  of  wood,  \"  thick. 
This  box  is  shown  orthographically  in  Fig.  20,  and  requires  two 
distinct  views  to  illustrate  it  properly.  The  upper  view,  or 
elevation,  shows  the  side  of  the  box  whose  outside  dimensions 
are  4"X12",  while  the  thickness  of  the  wood  is  indicated  by  the 


!& 


0 

o 

o 
o 

FIG.  20. 


FIG.  21. 


dotted  lines.  The  lower  view  is  called  the  plan,  and  is  obtained 
by  looking  down  into  the  inside  of  the  box.  It  is  thus  to 
be  remembered  that  the  two  views  are  due  to  two  distinct 
directions  of  vision  on  the  part  of  the  observer. 

As  another  example  of  this  mode  of  representation,  consider 
the  object  shown  in  Fig.  21.  It  is  here  a  rectangular  plate, 
5"X3"  and  I"  thick,  with  a  square  hole  in  its  centre.  The 
metal  around  the  square  hole  projects  \"  above  the  surface  of 
the  plate.  In  addition,  there  are  four  bolt  holes  which  enable 
the  part  to  be  secured  to  a  machine  with  bolts.  As  before, 
two  views  are  shown  with  the  necessary  dimensions  for  con- 
struction. 

25 


26  PARALLEL  PROJECTING-LINE  DRAWING 

302.  Theory  of  orthographic  projection.  Let  Fig.  22 
represent  an  oblique  projection  of  two  plane  surfaces  HH  and 
W,  at  right  angles  to  each  other.  For  mechanical  operations 
to  be  performed  later,  it  is  assumed  that  they  are  hinged  at 
their  intersection  so  that  both  planes  may  be  made  to  lie  as  one 
flat  surface,  instead  of  two  separate  surfaces  at  right  angles 
to  each  other.  The  plane  HH,  shown  horizontally,  is  the  hori- 
zontal plane  of  projection;  that  shown  vertically,  is  the  vertical 
plane  of  projection;  their  intersection  is  called  the  ground  line. 
The  two  planes,  taken  together,  are  known  as  the  principal  planes 


FIG.  22. 


of  projection.  The  object  is  the  4"X6"X12"  box  chosen  as 
an  illustration  in  Fig.  20.  The  drawing  on  the  horizontal  plane 
is  the  horizontal  projection;  while  that  on  the  vertical  plane  is 
the -vertical  projection. 

The  method  of  constructing  the  projection  consists  of  dropping 
perpendiculars  from  the  object  upon  the  planes  of  projection. 
Thus,  in  other  words,  the  projecting  lines  are  perpendicular  to 
the  plane  of  projection.  To  illustrate:  The  box  is  so  located 
in  space  that  the  bottom  of  it  is  parallel  to  the  horizontal  plane 
(Fig.  22)  and  the  4"X12"  side  is  parallel  to  the  vertical  plane. 
From  the  points  A,  B,  C,  and  D,  perpendiculars  are  drawn  to 
the  vertical  plane  and  the  points,  where  these  perpendiculars 
pierce  or  impinge  on  the  plane,  are  marked  a',  b';  c',  and  d',  to 


ORTHOGRAPHIC  PROJECTION  27 

correspond  with  the  similarly  lettered  points  on  the  object. 
By  joining  these  points  with  straight  lines,  to  correspond  with 
the  lines  on  the  object,  the  vertical  projection  is  completed, 
when  the  dotted  lines  showing  the  inside  of  the  box  are  added. 

Turning  to  the  projection  on  the  horizontal  plane,  it  is  seen 
that  A,  B,  E,  and  F  are  the  corners  of  the  box  in  space,  and 
that  perpendiculars  from  these  points  to  the  horizontal  plane 
determine  a,  b,  e,  and  f  as  the  horizontal  projection.  It  is 
assumed  that  the  observer  is  looking  down  on  the  horizontal 
plane  and  therefore  sees  the  inside  of  the  box;  these  lines  are 
hence  shown  in  full,  although  the  projecting  perpendiculars  are 
omitted  so  as  to  avoid  too  many  lines  in  the  construction. 

303.  Revolution  of  the  horizontal  plane.     It  is  manifestly 
impracticable  to  carry  two  planes  at  right  angles  to  each  other, 
each    containing    one    projection    of    an    object.     A    more    con- 
venient way  is  to  represent  both  projections  on  a  single    plane 
surface,  so  that  such  drawings  can  be  represented  on  a  flat  sheet 
of  paper.     The  evident  expedient,   in  this  case,   is  to  revolve 
the  horizontal  plane  about  the  ground  line  as  an  axis,  until  it 
coincides  with  the  plane  of  the  vertical  plane.     The  conventional 
direction  of  rotation  is  shown  by  the  arrow  in  Fig.  22,  and  to 
accomplish  this  coincidence,   a  90°   revolution  is  required.     In 
passing,  it  may  be  well  to  note  that  it  makes  no  difference  whether 
the  horizontal  plane  is  revolved  as  suggested,   or  whether  the 
vertical  plane  is  revolved  in  the  opposite  direction  into  coin- 
cidence with  the  horizontal  plane.     Both  accomplish  t>e  same 
purpose,  and  hence  either  method  will  answer  the  requirements. 

304.  Position  of  the  eye.     The  perpendicular  projecting  lines 
drawn  to  the  planes  of  projection  correspond  with  a  line  of  sight 
that  coincides  with  these  perpendiculars.     Each  point  found  en 
the  projection,  corresponds  to  a  new  position  of  the  eye.     All 
projecting  lines  to  one  plane  are  then  evidently  parallel  because 
they  are  all  perpendicular  to  the  same  plane.     As  two  projec- 
tions are  required,  two  general  directions  of  vision  are  necessary. 
That  for  the  horizontal  projection  requires  the  eye  above  that 
plane,  continually  directed  perpendicularly  against  it;    thus  the 
eye  is  continually  shifting  in  position,  although  the  direction  of 
vision  is  fixed.     Also,  the  vertical  projection  requires  that  the 
eye  be  directed  perpendicularly  against  it,  but  in  this  case,  the 


28 


PARALLEL  PROJECTING-LINE    DRAWING 


line  of  sight  is  perpendicular  to  that  required  for  the  horizontal 
projection. 

305.  Relation  of  size  of  object  to  size   of  projection. 

The  object  is  projected  on  the  planes  by  lines  perpendicular 
to  it.  If  the  plane  of  the  object  is  parallel  to  the  plane  of  pro- 
jection, then  the  projection  is  equal  to  the  object  in  magnitude. 
This  is  true  because  the  projecting  lines  form  a  right  prism  and 


FIG.  23. 

all  the  parallel  plane  sections  are  the  same  (compare  with  202). 
Fig.  23  gives  the  construction  of  the  projection  in  Fig.  21. 
ABCDa'b'c'd'  is  such  a  right  prism  because  the  plane  of  the  object 
is  parallel  to  the  plane  of  projection  and  the  projecting  lines 
are  perpendicular  to  the  plane  of  projection. 

306.  Location  of  object  with  respect  to  the  planes  of 
projection.  For  purposes  of  drawing,  the  location  of  the  object 
to  the  planes  of  projection  is  absolutely  immaterial.  In  fact, 


ORTHOGRAPHIC  PROJECTION  29 

the  draftsman  intuitively  makes  the  projections  and  puts  corre- 
sponding projections  as  close  as  is  necessary  to  economize  room 
on  the  sheet. 

307.  Location  of  projections  with  respect  to  each  other. 

In  Figs.  20  and  21,  the  vertical  projection  is  placed  directly 
above  the  horizontal  projection.  Reference  to  Fig.  23  will  show 
why  such  is  the  case.  When  the  horizontal  plane  is  revolved 
into  coincidence  with  the  vertical  plane,  the  point  d  will  describe 
the  arc  of  a  circle  dd"  *  which  is  a  quadrant;  d"  is  the  ultimate 
position  of  the  point  d  after  revolution,  and  must  be  on  a  line 
d'd"  which  is  perpendicular  to  the  original  position  of  the  hori- 
zontal plane.  So,  too,  every  point  of  the  horizontal  projection 
is  located  directly  under  the  corresponding  point  in  the  vertical 
projection,  and  the  scheme  for  finding  its  position  is  identical 
to  that  for  finding  d  at  d". 

308.  Dimensions  on  a  projection.   When  the  principal  planes 
of  the  object  are  turned  so  that  they  are  parallel  to  the  planes  of 
projection,   then  the  edges  will,  in  the  main,   be  perpendicular 
to  the  planes  of  projection.     In  Fig.  23,  DF  is  one  edge  of  the 
object  and  it  is  perpendicular  to  the  vertical  plane  of  projection 
W.     The  projection  of  this  line  is  d',   because  the  projecting 
perpendicular  from  any  point  on  DF  will  coincide  with  DF  itself. 
The  result  of  this  is  that  the  thickness  of  the  object  is  not  shown 
when  the  length  and  breadth  are  shown,  or,  in  other  words,  only 
two  of  the  three  principal  dimensions  are  shown  in  a  single  view. 
Thus,  another  view  is  required  to  show  the  thickness.     If  DC 
be  considered  a  length,  and  DF  a  thickness,  the  horizontal  pro^ 
jection  shows  both  as  dc  and  df.     The  vertical  projection  does 
not  show  the  thickness  DF  as  it  is  perpendicular  to  the  vertical 
plane  of  projection.     Hence,  in  reading  orthographic  projections, 
both  views  must  be  interpreted  simultaneously,  as  each  shows 
but  two  of  the  three  principal  dimensions  and  only  one  of  the 
three  is  common  to  both  projections. 

309.  Comparison    between    oblique    and    orthographic 
projections.     It  is  of  interest  here  to  show  wherein  the  ortho- 
graphic projection  differs  from  the  oblique.     When  the  plane 

*  d'  is  read  d  prime;  d"  is  read  d  second;  d"'  is  read  d  third;  and  so 
on. 


30 


PARALLEL  PROJECTING-LINE  DRAWING 


of  the  object  is  parallel  to  the  plane  of  projection,  the  projection 
on  that  plane  is  equal  to  the  object,  whether  it  is  projected 
orthographically  or  obliquely.  When  a  line  is  perpendicular  to 
the  plane  of  projection  its  extremities  have  two  distinct  pro- 
jections in  oblique  projection,  but  only  one  in  orthographic  pro- 
jection. This  latter  statement  means  simply  that  if  the  projecting 
lines  instead  of  being  oblique  to  the  plane  of  projection,  gradually 
assume  the  perpendicular  position,  the  two  projections  of  the 
extremities  of  any  line  approach  each  other  until  they  coincide 
when  the  projecting  lines  are  perpendicular.  Therefore,  in 
orthographic  projection,  the  third  dimension  vanishes  and  a 
new  view  must  be  made  in  addition  to  the  other,  in  order  to 
represent  a  solid. 

310.  Orthographic  projection  considered  a  shadow.     The 

horizontal  and  the  vertical  projections  may  be  considered  as 
shadows  on  their  respective  planes.  The  source  of  light  must 
be  such  that  the  rays  emanate  in  parallel  lines,  and  are  directed 
perpendicularly  to  the  planes  of  projection.  -Evidently,  the 
two  views  are  due  to  two  distinct  positions  of  the  source  of  light, 
one  whose  rays  are  perpendicular  to  the  horizontal  plane  while 
casting  the  horizontal  shadow,  and  the  other,  whose  rays  are 
perpendicular  to  the  vertical  plane  while  casting  the  vertical 
shadow. 

311.  Profile  plane.     Let  A,  in  Fig.   24,   be   the   horizontal 
projection   and  B,   the   vertical   projection  of   an  object.     The 

two  views  are  identical,  and  to 
one  unfamiliar  with  the  object, 
they  are  indefinite,  as  it  is  im- 
possible to  tell  whether  they  are 
projections  of  a  cylinder  or  of  a 
prism.  By  the  addition  of  either 
view  C  or  D,  it  is  at  once  appar- 
ent that  the  object  in  question 


is  a  circular    cylinder,  a    hole 
FIG.  24.  running    part  way   through  it 

and  with  one  end  square. 

Fig.  25  shows  how  this  profile  is  made.  As  customary,  the 
horizontal  and  vertical  planes  are  present  and  the  projection 
on  these  planes  should  now  require  no  further  mention.  A 


ORTHOGRAPHIC   PROJECTION 


31 


profile  plane  (or  end  plane  as  it  may  be  called)  is  shown  on  the 
far  side  of  the  object  and  is  a  plane  that  is  perpendicular  to 
both  the  horizontal  and  vertical  planes  (like  the  two  adjacent 


FIG.  25. 


walls  and  the  floor  of  a  room  meeting  in  one  corner).     A  series 
of  perpendiculars  is  dropped  from  the  object  upon  this  profile 


FIG.  26. 

plane,  as  shown  by  the  dotted  lines,  and  thus  the  side  view  is 
determined. 

312.  Location  of  profiles.  If  the  profile  view  is  to  show 
the  object  as  seen  from  the  left  side,  it  is  put  on  the  left  side  of 
the  drawing,  and  vice  versa.  Fig.  24  shows  two  profile  views 
located  in  accordance  with  this  direction.  Either  views  B  and  C 


32 


PARALLEL  PROJECTING-LINE  DRAWING 


or  B  and  D  completely  represent  the  object.  In  this  case, 
although  this  is  not  always  so,  the  horizontal  projection  is  not 
essential. 

Fig.  26  gives  still  another  illustration  of  an  object  that  is 
not  as  symmetrical  as  that  immediately  preceding.  The  illus- 
tration is  chosen  to  show  exactly  how  the  profile  planes  are 
revolved  into  the  vertical  plane,  if  the  vertical  plane  be  assumed 
as  the  plane  of  the  paper.  A  is  the  horizontal  and  B  the  vertical 
projection  of  the  object.  C  and  D  are  two  profiles,  drawn  against 
the  vertical  projection,  whereas  E  is  a  profile  drawn  against 
the  horizontal  projection.  Fig.  27  shows  a  plan  view  of  the 
vertical  and  two  profile  planes.  In  reading  this  drawing,  the 


FIG.  27. 

horizontal  plane  is  the  plane  of  the  paper,  while  the  vertical  plane 
is  seen  on  edge  and  is  shown  as  W,  as  are  also  the  left  and  right 
profile  planes  indicated  respectively  as  LL  and  RR. 

In  making  the  projection  on  the  horizontal  plane,  the  object 
is  above  the  plane  and  the  projecting  perpendiculars  are  dropped 
from  points  on  the  object  to  the  horizontal  plane,  which  in  this 
case  is  the  plane  of  the  paper.  The  construction  of  the  vertical 
projection  (that  on  W)  is  indicated  by  the  arrow  A.  The 
arrangement  here  shown  corresponds  to  the  views  A  and  B  in 
Fig.  26. 

When  making  the  profile  projections,  the  planes  are  assumed 
as  transparent,  and  are  located  between  the  object  and  the 
observer.  As  the  observer  traces  the  outline  on  these  profile 
planes,  point  by  point,  each  ray  being  perpendicular  to  the  plane, 
the  resultant  picture  so  drawn  becomes  the  required  projection. 
If,  then,  the  planes  LL  and  RR  be  revolved  in  the  direction  of  the 


ORTHOGRAPHIC   PROJECTION 


33 


arrows  until  they  coincide  with  the  vertical  plane,  and  then  the 
vertical  plane  be  further  revolved  into  the  plane  of  the  paper, 
the  final  result  will  be  that  of  Fig.  26  with  view  E  omitted. 

View  E  is  a  profile  drawn  against  the  horizontal  projection 
and  is  shown  on  the  left  because  it  is  the  projection  on  the  profile 
plane  LL.  It  has  been  revolved  into  the  horizontal  plane,  by 
revolving  the  profile  plane  so  that  the  upper  part  of  the  plane 
moves  toward  the  object  into  coincidence  with  the  horizontal 
plane. 

Fig.  26  has  more  views  than  are  necessary  to  illustrate  the 
object  completely.  In  practice,  all  would  not  be  drawn,  their 
presence  here  is  necessary  only  to  show  the  method. 

313.  Section  plane.  The  addition  of  dotted  lines  to  the 
drawing  of  complicated  objects  is  unsatisfactory  at  times  on 
account  of  the  resultant  confusion  of  lines.  This  difficulty  can 
be  overcome  by  cutting  the  object  by  planes,  known  as  section 
planes.  The  solid  material  when 
so  exposed  is  sectioned  or  cross- 
hatched  by  drawing  a  series 
of  equidistant  lines  over  the 
exposed  area.  A  convenient 
mnemonic  in  this  connection  is 
to  assume  that  the  cut  is  made 
by  a  saw  and  that  the  resultant 
tooth  marks  represent  the  sec- 
tion lines.  Fig.  28  shows  what 

is  known  as  a  stuffing  box  on  a  steam  engine.  This  is  a  special 
case  where  but  one  projection  is  shown  in  section  and  one  profile. 
The  left-hand  view  might  have  been  shown  as  an  outside  view, 
but  the  interior  lines  would  then  have  been  shown  dotted.  As 
it  is,  the  object  is  cut  by  the  plane  ab  and  this  half  portion  is 
shown  to  the  left,  sectioned  of  course,  because  the  cut  is  not 
actual. 

Another  example  is  seen  in  Fig.  29,  where  a  fly-wheel  is 
represented  in  much  the  same  way  as  in  the  illustration  in  Fig. 
28.  It  differs  somewhat  from  that  immediately  preceding  in 
so  far  as  the  two  views  do  not  have  the  theoretical  relation. 
Were  the  wheel  actually  cut  by  the  plane  ab  then  the  arms 
(spokes)  shown  in  the  profile  would  have  to  be  sectioned.  As 


FIG.  28. 


34  PARALLEL  PROJECTING-LINE  DRAWING 

shown,  however,  the  arms  appear  in  full  as  though  the  section 
plane  passed  through  the  wheel  a  short  distance  ahead  of  the 
spokes.  The  convention  is  introduced  for  a  double  purpose: 
In  the  first  place  it  avoids  peculiar  projections  as  that  for  the 
plane  cd  for  instance,  where  the  spokes  would  be  foreshortened 
because  they  incline  to  the  plane  of  projection.  In  the  second 
place,  the  sectioning  of  the  spokes  is  the  conventional  method 
of  showing  a  band  wheel,*  that  is,  a  wheel  with  a  solid  web, 
or,  in  other  words,  without  spokes.  Hence,  it  appears  that 
although  it  may  not  seem  like  a  rational  method  of  drawing, 
still  the  attending  advantages  are  such  as  make  it  a  general 
custom.  The  mechanics  who  use  the  drawings  understand  this, 
and  therefore  it  becomes  common  practice. 


FIG.  29. 

\ 

Many  more  examples  could  be  added,  but  they  would  be 
too  complicated  to  be  of  illustrative  value.  It  may  be  said,  that, 
in  some  cases,  six  or  more  sections  may  be  made  to  illustrate 
the  object  completely.  They  are  located  anywhere  on  the  draw- 
ing and  properly  indicated,  similar  to  cd  in  Fig.  29.  It  may  also  be 
mentioned,  that  in  cases  like  that  of  the  fly-wheel,  the  shaft 
is  not  cut  by  the  section  plane  but  is  shown  in  full  as  it  appears 
in  Fig.  29. 

314.  Supplementary  plane.  Fig.  30  shows  a  Y  fitting  used 
in  pipe  work  for  conveying  steam,  water,  etc.,  and  consists  of 
a  hollow  cylindrical  shell  terminating  in  two  flanges,  one  at 

*  Fig.  43  is  an  example  of  a  band  wheel. 


ORTHOGRAPHIC  PROJECTION 


35 


each  end.     From  this  shell  there  emerges  another  shell  (in  this 

case,  smaller  in  diameter),  also  terminating  in  a  flange.     A  is  the 

Y  fitting  proper;    B  is  the  end  view  of  one  flange,  showing  the 

bolt    holes    for    fastening  to   a 

mating     flange     on    the     next 

piece  of  pipe,  not  shown.     The 

view    B    shows    only    the    one 

flange   that  is    represented    by 

a    circle,    because    the    profile 

plane   is    chosen  so    as    to   be 

parallel  to  that  flange.     If  the 

flange   at    C    be    projected    on 

this    same    profile    plane,    it  FIG.  30. 

would    appear    as    an    ellipse, 

and,  as  such,  could  not  be  drawn  with  the  same  facility  as  a 

circle,     Here,  then,  is  an  opportunity  to  locate  another  plane, 


FIG.  31. 


called  a  supplementary  plane,  parallel  to  the  flange  at  C.  The 
projection  of  C  on  this  supplementary  plane  will  be  a  circle 
and  is  therefore  readily  drawn  with  a  compass. 


36 


PARALLEL  PROJECTING-LINE   DRAWING 


Only  one-half  of  the  actual  circle  may  be  shown  if  desired 
so  as  to  save  time,  space,  or  both  time  and  space.  The  bolt 
holes  are  shown  in  the  supplementary  view  at  C  just  the  same 
as  in  view  B.  Either  plane  may  be  considered  supplementary 
to  the  other;  on  neither  plane  is  the  entire  projection  made, 
because  the  object  is  of  so  simple  a  character.  To  avoid  the 
possibility  of  any  error  arising,  the  supplementary  project' en 
is,  if  it  is  at  all  possible,  located  near  the  part  to  be  illustrated. 
If,  for  any  reason,  this  view  cannot  be  so  located,  a  note  indicating 
the  proper  position  of  the  view  is  added  to  the  drawing. 

315.  Angles  of  projection.  Up  to  the  present  point,  no 
attention  has  been  devoted  to  the  angles  in  which  the  pro- 


0 

i  i 
^N 

0 

0 

1 
1 
1 
1 

C 

r^ 

o 

—  1 

1 

1 

0 

I  i 

o 

1          1 

1st  Angle        2nd  Angle       3rd  Angle 
FIG.  32. 


4th  Angle 


jections  were  made.  As  will  soon  appear,  the  examples  so  far 
chosen  were  all  in  the  first  angle.  Fig.  31  shows  two  planes 
HH  and  W,  intersecting  at  right  angles  to  each  other.  The 
planes  form  four  dihedral  angles,  numbered  consecutively  in 
a  counter-clockwise  manner  as  indicated.  The  same  object  is 
shown  in  all  four  angles,  as  are  also  the  projections  on  the 
planes  of  projection,  thus  making  four  distinct  projections. 

316.  Location  of  observer  in    constructing  projections. 

The  eye  is  always  located  above  the  horizontal  plane  in  making 
any  horizontal  projection.  That  is,  for  objects  in  the  first  and 
second  angles,  the  object  is  between  the  plane  and  the  observer ; 
for  objects  in  the  third  and  fourth  angles  the  plane  is  between 


ORTHOGRAPHIC  PROJECTION  37 

the  object  and  the  observer.  While  constructing  the  vertical 
projections  the  eye  is  always  located  in  front  of  the  vertical  plane. 
That  is,  for  objects  in  the  first  or  fourth  angles,  the  object 
is  between  the  plane  and  the  observer ;  but  in  the  second  and 
third  angles,  the  plane  is  between  the  object  and  the  observer. 
This  latter  means  simply  that  the  observer  stands  to  the  right  of 
this  vertical  plane  W  and  views  it  so  that  the  line  of  sight  is 
always  perpendicular  to  the  plane  of  projection. 

317.  Application   of   angles   of    projection   to    drawing. 

If  the  horizontal  plane  be  revolved  about  the  ground  line  XY 
as  indicated  by  the  arrows,  until  it  coincides  with  the  vertical 
plane  of  projection,  it  will  be  seen  that  that  portion  of  the  hori- 
zontal plane  in  front  of  the  vertical  plane  will  fall  below  the 
ground  line,  whereas  that  portion  of  the  horizontal  plane  behind 
the  vertical  plane  will  rise  above  the  ground  line. 

Supposing  that  in  each  position  of  the  object  in  all  four 
angles,  the  projections  were  made  by  dropping  the  customary 
perpendiculars  to  the  plane  of  projection,  and  in  addition,  that 
the  revolution  of  the  horizontal  plane  is  accomplished,  then 
the  resultant  state  of  affairs  will  be  as  shown  in  Fig.  32.  For 
purposes  of  illustration,  the  ground  line  XY  is  drawn  although 
it  is  never  used  in  the  actual  drawing  of  objects.*  Also,  the 
object  has  been  purposely  so  located  with  respect  to  the  planes 
that  the  second  and  fourth  angle  projections  overlap.  Mani- 
festly the  second  and  fourth  angles  cannot  be  used  in  drawing 
if  we  wish  to  be  technically  correct.  It  may  be  possible  so  to 
locate  the  object  in  the  second  and  fourth  angles,  by  simply 
changing  the  distance  from  one  plane  or  the  other,  that  the 
two  projections  do  not  conflict,  but  a  little  study  will  show  that 
the  case  falls  either  under  first  or  third  angle  projection,  depending 
upon  whether  the  vertical  projection  is  above  the  horizontal 
projection,  or,  below  it. 

It  will  be  seen  that  in  the  first  angle  of  projection,  the  plan 
is  below  and  the  elevation  is  above;  whereas  in  the  third  angle 
of  projection  the  condition  is  reversed,  that  is,  the  plan  is  above 
and  the  elevation  is  below.  Strictly  speaking,  the  profile,  section 
and  supplementary  planes,  have  nothing  to  do  with  the  angle 

*  When  lines,  points,  and  planes  are  to  be  represented  orthographically, 
the  ground  line  becomes  a  necessary  adjunct. 


38 


PARALLEL  PEOJECTING-LINE  DRAWING 


of  projection,  but  it  is  quite  possible  to  take  a  single  projection 
with  its  profile,  and  locate  it  so  that  it  corresponds  to  a  third 
angle  projection.  Thus,  there  appears  a  certain  looseness  in 
the  application  of  these  principles.  In  general,  the  third  angle 
of  projection  is  used  more  than  any  other  *  as  the  larger  number 


o 

-1  — 

i 

0 

1st  Angle 


3rd  Angle 


FIG.  33. 


of  mechanics  are  familiar  with  the  reading  of  drawings  in  this 
ftngle.  Fig.  33  shows  the  same  object  in  the  first  and  third 
angles  of  projection.  A  profile,  or  end  view,  is  also  attached 
to  each,  thus  making  a  complete,  though  simple  illustration. 

318.  Commercial  application  of  orthographic  projection. 

Orthographic  projection  is  by  far  the  most  important  method 
of  making  drawings  for  engineering  purposes.  Other  types  of 
drawing  have  certain  advantages,  but,  in  general,  they  are  limited 
to  showing  simple  objects,  made  up  principally  of  straight  lines. 

Some  experience  is  required  in  reading  orthographic  projections 
because  two  or  more  views  must  be  interpreted  simultaneously. 
This  experience  is  readily  acquired  by  practice,  both  in  making 
drawings  and  in  reading  the  drawings  of  others. 

To  compensate  for  the  more  difficult  interpretation  of  this 
type  of  drawing,  there  are  inherent  advantages,  which  permit 
the  representation  of  any  object,  if  it  has  some  well  defined 
shape.  By  the  aid  of  sections,  profiles,  and  supplementary 
planes,  any  side  of  a  regular  body  can  be  illustrated  at  will, 
and  further  than  this,  the  curves  are  shown  with  such  peculiarity 
as  characterizes  them.  Bodies,  such  as  a  lump  of  coal,  or  a  spade 

*  The  third  angle  of  projection  should  b6  used  in  preference  to  the  first 
because  the  profile,  section,  and  supplementary  planes  conform  to  third 
angle  projection. 


ORTHOGRAPHIC  PROJECTION 


39 


full  of  earth,  are  considered  shapeless,  and  are  never  used  in 
engineering  construction.  Even  these  can  be  represented  ortho- 
graphically,  however,  although  it  is  quite  difficult  to  draw  on 
the  imagination  in  such  cases. 


QUESTIONS  ON  CHAPTER  III 

1.  What  are  the  principal  planes  of  projection?    Name  them. 

2.  What  is  the  ground  line? 

3.  What  angles  do  the  orthographic  projecting  lines  make  with  the 

plane  of  projection? 


FIG.  3A. 


4.  What  is  the  horizontal  projection? 

5.  What  is  the  vertical  projection? 

6.  Why  is  the  horizontal  plane  revolved  90°  after  making  the  pro- 

jections? 


FIG.  ZB. 

7.  Could  the  vertical  plane  be  revolved  instead  of  the  horizontal  plane? 

8.  Make  a  sketch  of  the  planes  of  projection  and  show  by  arrow  how 

the  revolution  of  the  planes  is  accomplished. 

9.  How  is  the  eye  of  the  observer  directed  in  making  an  orthographic 

projection? 


40 


PARALLEL  PROJECTING-LINE  DRAWING 


10.  What  is  the  general  angular  relation  between  the  projecting  lines 

to  the  horizontal  plane  and  the  vertical  plane  of  projection? 

11.  Why  is  the  projection  of  the  same  size  as  the  object? 

12.  Why  is  the  ground  line  omitted  in  making  orthographic  projections? 

13.  Why  are  corresponding  projections  located  directly  over  each  other? 

Show  by  diagram. 

14.  Why  does  an  orthographic  projection  show  only  two  of  the  three 

principal  dimensions  of  the  object? 


"1  " 


FIG.  3C. 

15.  Why  must  the  two  views  of  an  orthographic  projection  be  interpreted 

simultaneously? 

16.  Compare  the  direction  of  the  projecting  lines  of  oblique  projection 

with  those  of  orthographic  projection. 

17.  Show  how  the  source  of  light  must  be  located  in  order  that  the 

shadow  should  correspond  to  a  true  projection. 


FIG.  3D. 

18.  To  cast  the  horizontal  and  vertical  shadows,  is  it  necessary  to  have 

two  distinct  positions  for  the  source  of  light? 

19.  What  is  a  profile  plane? 

20.  How  is  the  profile  plane  located  with  respect  to  the  principal  planes 

of  projection? 

21.  How  is  the  profile  plane  revolved  into  the  plane  of  the  drawing? 

Show  by  diagram. 


ORTHOGRAPHIC   PROJECTION 


41 


22.  How  are  the  profiles  located  with  respect  to  the  main  projection? 

Show  by  some  simple  sketch. 

23.  What  is  a  section  plane? 

24.  When  is  a  section  plane  desirable? 

25.  How  is  the  section  constructed? 


TIG.  3F. 


26.  How  is  the  section  located  with  respect  to  the  main  projection? 

27.  Show  a  simple  case  where  only  one  projection  and  one  section  com- 

pletely determine  an  object. 


FIG.  3F. 

28.  What  is  a  supplementary  plane? 

29.  When  is  it  desirable  to  use  a  supplementary  plane? 

30.  How  is  the  supplementary  projection  located  with  respect  to  the 

main  projection? 


42 


PARALLEL  PROJECTING-LINE  DRAWING 


31.  Show  why  the  profile  plane  is  a  special  case  of  the  supplementary 

plane. 

32.  Make  a  diagram  showing  the  four  angles  of  projection  and  show 

how  they  are  numbered. 


o 

o 

o 

0 

FIG.  3#. 


FIG.  3G. 


FIG.  37. 


33.  How  is  the  revolution  of  the  planes  accomplished  so  as  to  bring 

them  into  coincidence? 

34.  How  is  the  observer  located  in  making  first  angle  projections? 


ORTHOGRAPHIC  PROJECTION 


43 


35.  How  is  the  observer  located  in  making  second  angle  projections? 

36.  How  is  the  observer  located  in  making  third  angle  projections? 

37.  How  is  the  observer  located  in  making  fourth  angle  projections? 

38.  Why   are   only  the   first   and  third   angles   of  projection    used  in 

drawing? 

39.  How  does  the  third  angle  differ  from  the  first  angle  projection? 

Show  by  sketch. 

40.  Why  is  the  third  angle  to  be  preferred? 


FIG.  3J. 


41.  Why  is  it  more  difficult  to  read  orthographic  projections  than  oblique 

projections? 

42.  What  distinct  advantages  has  orthographic  projection  over  oblique 

projection? 


FIG. 


FIG.  3L. 


43.  Make  a  complete  working  drawing  of  3-A  and  show  one  view  in 

section.    Assume  suitable  dimensions. 

NOTE:    A  working  drawing  is  a  drawing,  completely  dimensioned, 
with  all  necessary  views  for  construction  purposes. 

44.  Make  a  complete  working  drawing  of  3-B  and  show  one  view  in 

section.    Assume  suitable  dimensions. 

45.  Make  a  complete  working  drawing  of  3-C  and  show  one  view  in 

section.     Assume  suitable  dimensions. 

46.  Make  a  complete  working  drawing  of  3-D  and  show  one. view  in 

section.    Assume  suitable  dimensions. 


44  PARALLEL  PROJECTING-LINE  DRAWING 

47.  Make  a  complete  working  drawing  of  3-E  and  show  the  profile  on 

the  left,  to  the  top  view.     Assume  suitable  dimensions. 

48.  Make  a  complete  working  drawing  of  3-F  and  show  the  profile  on 

the  left,  to  the  top  view.     Assume  suitable  dimensions. 

49.  Make  a  complete  working  drawing  of  3-G  and  show  the  lower  view 

as  a  first  angle  projection.     Assume  suitable  dimensions. 

50.  Rearrange  3-G  and  show  three  views  in  third  angle  projection. 

51.  Make  a  section  of  3-H  through  the  web  and  observe  that  the  web 

is  not  sectioned.     Assume  suitable  dimensions. 

52.  Make  a  supplementary  view  of  the  45°  ell  shown  in  3-1.    Assume 

suitable  dimensions. 

53.  Make  a  supplementary  view  of  3-J.    Assume  suitable  dimensions. 

54.  Make  three  views  of  3-K  in  third  angle  projection.    Assume  suitable 

dimensions. 

55.  Make  three  views  of  3-L  in  third  angle  projection.    Assume  suitable 

dimensions. 


CHAPTER  IV 
AXONOMETRIC  PROJECTION 

401.  Nature  of  isometric  projections.  Consider  three  lines 
intersecting  at  a  point  and  make  the  angles  between  each  pair 
of  lines  equal  to  120°;  the  three  angles  will  then  total  360°  which 
is  the  total  angle  about  a  point.  If  on  one  of  these  lines,  or  lines 
parallel  thereto,  lengths  are  laid  off,  on  the  other  line,  or  lines 
parallel  thereto,  breadths  are  laid  off,  and  on  the  remaining  line, 
or  lines  parallel  thereto,  thicknesses  are  laid  off,  it  seems  quite 
reasonable  that  a  method  may  be  devised  whereby  the  three 
principal  dimensions  can  be  plotted  so  as  to  represent  objects 
in  a  single  view.  This  method  when  carried  out  to  completion 
results  in  an  isometric  projection. 

Let,  in  Fig.  34,  OA,  OB,  and  OC  be  the  three  lines,  drawn 
as  directed,  so  that  the  angles  between  them  are  120°.  Suppose 
it  is  desired  to  draw  a  box 

4"X6"X12",  made  of  wood  *> 

\"  thick.  Lay  off  12"  on  OB 
(to  any  convenient  scale), 
6"  on  OA  and  4"  on  OC. 
From  A,  draw  a  line  AD 
parallel  to  OB  and  AE  par- 
allel to  OC;  also,  from  C 
draw  CE,  parallel  to  OA  and 
CF,  parallel  to  OB.  The 
thickness  of  the  wood  is  to 
be  added  and  the  direction 
in  which  this  is  laid  off  is 
indicated  by  the  direction 

of  the  corresponding  dimension  line.  In  the  drawing,  the  line 
OC  was  purposely  chosen  vertical  so  that  OA  and  OB  may  be 
readily  drawn  with  a  60°  triangle.  The  other  lines,  added  to 
Indicate  the  construction,  are  self-explanatory. 


FIG.  34. 


46 


PARALLEL  PROJECTING-LINE  DRAWING 


This,  then,  is  an  isometric  projection,  and,  as  may  be  noted, 
is  a  rapid  method  of  representing  objects  in  a  single  view.  Com- 
parison may  be  made  with  Figs.  1  and  20,  which  show  the  same 
box  in  oblique  and  orthographic  projection,  respectively. 

402.  Theory  of  isometric  projection.  Let  Fig.  35,  represent 
a  cube  shown  on  the  left  side  of  a  transparent  plane  W.  If 
the  observer,  located  on  the  right,  projects  this  cube  ortho- 
graphically  (projecting  lines  perpendicular  to  the  plane)  on  the 
plane  W,  and  at  the  same  time  has  the  cube  turned  so  that  each 
of  the  three  visible  faces  is  projected  equally  on  the  plane,  the 


resultant  projection  is  an  isometric  projection  of  the  cube  on 
the  plane  W. 

The  illustration  on  the  right  of  Fig.  35  shows  how  this  cube 
appears  when  orthographically  projected.  OA,  OB,  and  OC 
are  called  the  isometric  axes.  As  each  face  of  the  cube  is  initially 
equal  to  the  other  faces,  and  as  each  edge  is  also  equal,  then, 
with  equal  inclination  of  the  three  faces,  their  projections  are 
equal.  Hence,  the  three  angles  are  each  equal  to  120°  and  the 
three  isometric  axes  are  equal  in  length.  To  use  these  axes 
for  drawing  purposes  merely  requires  that  all  dimensions  of 
one  kind  (lengths,  for  instance)  be  laid  off  on  any  one  line,  or 
lines  parallel  thereto,  and  that  this  process  be  observed  for  the 
three  principal  dimensions. 

Isometric  projection,  is,  therefore,  a  special  case  of  ortho- 
graphic projection;  because,  in  its  conception,  the  principal  planes 


AXOXOMETRIC  PROJECTION 


47 


of  the  object  are  inclined  to  the  plane  of  projection.  As  solids 
are  thereby  represented  in  a  single  view,  only  one  projection 
is  necessary. 

403.  Isometric  projection  and  isometric  drawing.  When 
a  line  is  inclined  to  the  plane  of  projection,  the  orthographic 
projection  of  the  line  is  shorter  than  that  of  the  line  itself,  for,  if 
the  degree  of  inclination  continue,  the  line  will  eventually  become 
perpendicular  to  the  plane  and  will  then  be  projected  as  a  point. 
Thus,  in  Fig.  35,  OA,  OB,  and  OC  are  drawn  shorter  than  the 
actual  edges  of  the  cube,  and  the  projection  on  the  right  is  a  true 
isometric  projection.  In  the  application  of  this  mode  of  pro- 


FIG.  36 


jection,  however,  it  is  easier  to  lay  off  the  actual  distance  (01 
any  proportion  of  it)  rather  than  this  foreshortened  distance, 
If  commercial  scales  are  used  in  laying  out  the  drawing  instead 
of  the  true  isometric  projections,  then  it  is  called  an  Isometric 
Drawing. 

The  distinction  between  the  two  is  a  very  fine  one,  since, 
if  the  ratio  of  foreshortening  *  be  used  as  a  scale  to  which  the 
drawing  is  made,  then  it  is  possible  by  a  simple  statement,  to 
change  from  isometric  drawing  to  isometric  projection.  The 
commercial  name  will  be  followed  and  they  will  be  called  isometric 
drawings,  always  bearing  in  mind  that  the  distinction  means 
little. 

*  This  ratio  of  the  actual  dimension  to  the  true  projected  dimension  is 
100  :  83  and  may  be  computed  by  trigonometry. 


48 


PARALLEL  PROJECTING-LINE  DRAWING 


404.  Direction  of  axes.     It  is  usual  to  assume  one  of  the 
axes  as  either    horizontal,  or  vertical,   as  under  these  circum- 
stances, the  other  two  can  be  drawn  with  the  60°  triangles  which 
are  standard  appliances  in  the  drawing  room.     Fig.  36  shows 
a  wood  block  in  several  positions,  each  of  which  is  an  isometric 
drawing;    here,  the  location  of  the  observer  with  respect  to  the 
block  is  at  once  apparent. 

405.  Isometric  projection  of  circles.    As  no  line  is  shown 
in  its  true  length  when  it  lies  in  the  faces  of  the  cube  and  is 
projected  isometrically,  then,  also,  no  curve  that  lies  in  these 
faces  can  be  shown  in  its  true  length  and,  therefore,  in  its  true 
shape.     This  is  due  to  the  foreshortening  caused  by  the  inclina- 


FIG.  37 

tion  of  the  plane  of  the  three  principal  dimensions  to  the  plane 
of  projection. 

Fig.  37  shows  two  cubes,  the  one  on  the  left  appears  as  a  single 
square  because  it  is  an  orthographic  projection  and  this  plane 
was  parallel  to  the  plane  of  projection.  On  the  right,  an  iso- 
metric drawing  is  shown.  In  the  orthographic  projection  on 
the  left,  a  circle  is  inscribed  in  the  face  of  the  cube.  The  circle, 
so  drawn,  is  tangent  to  the  sides  of  the  square  at  points  midway 
between  the  extremities  of  the  lines.  When  this  square  and  its 
inscribed  circle  is  shown  isometrically,  the  points  of  tangency 
do  not  change,  but,  as  the  square  is  projected  as  a  rhombus, 
the  circle  is  projected  as  an  ellipse,  and  is  a  smooth  curve  that 
is  tangent  at  mid-points  of  the  sides  of  the  rhombus.  A  rapid, 


AXONOMETRIC  PROJECTION 


49 


though  approximate  method  of  drawing  an  ellipse  *  is  shown 
in  the  upper  face  of  the  isometric  cube.  The  major  and  the 
minor  axes  of  the  ellipse  can  be  laid  off  accurately  by  drawing 
the  diagonals  eg  and  fh  (the  notation  being  alike  in  both  views) ; 
a  tangent  to  the  circle  is  perpendicular  to  the  radius,  and,  for 
points  on  the  diagonals,  this  tangent  is  shown  as  mn  (all  four 
are  alike  because  they  are  equal).  Showing  mn  isometrically 
means  that  the  gm  =  gn  in  one  view  is  equal  to  the  gm  =  gn  in 
the  other  view;  the  hm  =  hn  in  one  view  is  equal  to  the  hm  =  hn 
in  the  other  view,  and  so  on,  for  the  four  possible  tangents  at 
the  diagonals. 

406.  Isometric  projection  of  inclined  lines  and  angles. 

Suppose  it  is  desired  to  locate  the  centre  of  a  hole  in  one  of  the 

faces  of  a  cube  as  at  h  (Fig.  38). 

The  hole  is   12"   back  from   the 

point  f,  on  the  line  fm,  and  6"  up 

from    the   point   m;    hfm  is  the 

isometric  representation  of  a  right 

triangle  whose    legs    are    6"    and 

12".      It   will    be   observed    that 

none   of   the  right  angles   of   the 

cube  are  shown  as  such,  therefore, 

the  angle  hfm  is  not  the  true  angle 

corresponding  to  these  dimensions; 

and,  hence,  it  cannot  be  measured 

with  a  protractor  in  the  ordinary 

way. 

The  point  k  is  located  in  a  similar  manner  on  the  top  of  the 
cube,  while  ank  is  the  isometric  drawing  of  a  right  triangle  whose 
legs  are  9"  and  13".  A  diagonal  ed  of  the  cube  is  also  shown. 

The  general  method  of  drawing  any  line  in  space  is  to  plot  the 
three  components  of  the  line.  For  instance,  the  point  d  is 
located  with  respect  to  the  point  e,  by  first  laying  off  ef,  then 
fg,  and  finally  gd. 

407.  Isometric  graduation  of   a  circle,  t    If  the  plane  of 
a  circle  is  parallel  to  the  plane  of  projection,  the  projection  is 

*  The  exact  method  of  constructing  an  ellipse  will  be  found  in  any  text- 
book on  geometry. 

f  This  method  is  also  applicable  to  oblique  projection. 


FIG.  38. 


50  PARALLEL  PROJECTING-LINE  DRAWING 

equal  to  the  circle  itself  and,  hence,  it  can  be  drawn  with  a  compass 
to  the  required  radius.  Any  angle  is  then  shown  in  its  true 
size  and  thus  the  protractor  can  be  applied  in  its  graduation. 

When  circles  are  shown  isometrically,  however,  their  pro- 
jections are  ellipses,  and  the  protractor  graduation  is  applicable 
no  longer.  In  the  upper  face  of  the  cube,  shown  isometrically 
in  Fig.  39,  the  circle  is  shown  as  an  ellipse  whose  major  and 
minor  axes  are  respectively  horizontal  and  vertical.  If,  on  the 
major  axis  ab,  a  semicircle  is  drawn,  and  graduated  by  laying 


FIG.  39. 

off  angles  at  30°  intervals,  the  points  CDEF  and  G  are  obtained 
making  six  angles  at  30°,  or  a  total  of  180°,  the  angular  measure 
of  a  semicircle.  If,  then,  the  plane  of  the  circle  is  rotated  about 
ab  as  an  axis,  until  the  point  E  coincides  with  e,  each  point  of 
division  on  the  semicircle  will  find  itself  on  the  similarly  lettered 
point  of  the  ellipse;  because,  then,  the  plane  of  the  semicircle 
coincides  with  the  plane  of  the  upper  face  of  the  cube.  Therefore, 
aoc,  cod,  doe,  etc.,  are  30°  angles,  shown  isometrically. 

In  the  side  face  of  the  same  cube  are  shown  two  methods  of 
laying  off  45°  angles.     The  fact  that  both  methods  locate  the 


AXONOMETRIC  PROJECTION 


51 


same  points  tends  to  show  that  either  method,  alone,  is  applicable. 
The  lettering  is  such  that  the  construction  should  be  clear  without 
further  explanation. 

408.  Examples  of  isometric  drawing.  As  an  example,  the 
steps  in  the  drawing  of  a  wooden  horse,  shown  in  Fig.  40  and 
used  in  the  building  trades  for  supporting  platforms  and  the 
like,  will  be  followed.  The  making  and  the  subsequent  interpre- 


FIG.  40. 

tation  of  drawings  of  this  kind  is  facilitated  by  the  introduction 
of  bounding  figures  of  simple  shape.  In  the  example  chosen, 
the  horse  is  bounded  by  a  rectangular  prism.  The  attached 
dimensions  show  the  necessary  slopes  and  may  be  introduced 
so  as  to  replace  the  bounding  figure.  There  is  nothing  new  in 
this  drawing  and  therefore  the  description  will  not  be  needlessly 
exhaustive.  The  bounding  figure,  when  appended  to  a  drawing 
like  that  of  Fig.  40,  helps  to  emphasize  the  slopes  of  the  various 
lines.  In  general,  the  bounding  figure  should  be  removed  on 
completion  of  the  drawing. 


52 


PARALLEL  PROJECTING-LINE  DRAWING 


Fig.  41  shows  a  toothed  wheel.  The  plane  of  the  circles 
corresponds  to  the  plane  of  the  top  face  of  a  cube.  To  construct 
it,  it  is  necessary  to  lay  out  the  prism  first  and  then  to  insert  the 
ellipses.  The  graduation  of  the  circle  is  similar  to  that  indicated 
in  Art.  407. 


TIG.  41. 


A  lever  is  shown  in  Fig.  42.  The  centre  line  ab  lies 
in  the  top  face  of  the  lever  proper.  Circumscribing  prisms 
determine  the  ends  of  the  lever  and  also  the  projecting  cylin- 
drical ends. 


FIG.  42. 


A  gas-engine  fly-wheel  is  illustrated  in  Fig.  43.  Here  the 
fly-wheel  is  shown  so  that  the  planes  of  the  circles  correspond 
to  one  of  the  side  faces  of  a  cube.  The  wheel  has  a  solid  web 
(without  spokes)  and  a  quarter  of  it  is  removed  and  shown  in 
section. 


AXONOMETRIC  PROJECTION  53 


PARALLEL  PROJECTIXG-LINE  DRAWING 


Fig.  44  represents  a  bell-crank.  Again  the  scheme  of  using 
base-lines  in  connection  with  circumscribing  prisms  is  shown. 
This  view  should  be  compared  with  that  given  in  Fig.  17. 

409.  Dimetric  projection  and 
dimetric  drawing.  Let  Fig.  45  be 
the  isometric  projection  of  a  cube, 
with  the  invisible  edges  shown  dotted. 
A  disturbing  symmetry  of  the  lines 
is  at  once  apparent.  This  objection 
becomes  serious  when  applied  to 
drawing  if  the  objects  are  cubical,  or 
nearly  so. 

The  foregoing  difficulty  may  be 
partially  overcome  by  turning  the 
cube  so  that  only  two  faces  are 
projected  equally  and  the  remaining 

face  may  be  larger  or  smaller  at  pleasure.  Fig.  46  shows 
this  condition  represented.  The  angle  aoc  is  larger  than  either 
the  angles  aob  or  cob,  the  latter  two  (aob  and  cob)  being 


FIG.  45. 


equal.  The  faces  A  and  B  are  projected  equally,  whereas  C 
is  smaller  in  this  particular  case,  though  it  need  not  be.  The 
illustration  as  shown  on  the  right  of  Fig.  46  is  a  dimetric  pro- 
jection and,  as  such,  the  same  scale  is  applied  to  the  axes  oa  and 


AXONOMETRIC  PROJECTION 


55 


oc  because  they  are  projected  equally.  The  axis  ob  is  longer, 
since  the  corresponding  edge  is  more  nearly  parallel  to  the  plane 
of  projection  W  than  either  oa  or  oc.  Hence,  to  be  theoretically 
correct,  a  different  scale  must  be  used  on  the  axis  ob. 

When  dimetric  projection  is  to  be  commercially  applied, 
confusion  may  result  from  the  use  of  two  distinct  scales.  If 
one  scale  is  used  on  all  three  axes  the  combination  becomes  a 
dimetric  drawing.  Hence,  a  dimetric  drawing  differs  from  an 
isometric  drawing,  in  so  far,  as  two  of  three  angles  are  equal 
to  each  other  for  the  dimetric  drawing;  whereas,  in  isometric 
drawing,  all  three  are  equal,  that  is,  the  axes  are  120° 
apart. 

What  is  true  of  the  direction  of  the  axes  in  isometric  drawing 
is  equally  true  here,  that  is,  the  angular  relation  between  the 
axes,  alone,  determines  the  type  of  projection,  the  direction  of 
any  one  is  entirely  arbitrary. 

Dimetric  drawings  do  not  entirely  remove  the  objectionable 
symmetry,  yet  they  find  some  use  in  practice,  although  they 
present  no  distinct  advantage  over  any  other  type. 

410.  Trimetric  projection  and  trimetric  drawing.     Fig. 
47  shows  a  cube  which  is  held   in  such  a  position  that   the 
orthographic    projection    of    it    will 
result  in   the   unequal    projection   of 
the  three  visible  faces.      This,   then, 
becomes  a  trimetric  projection. 

To  make  true  projections,  three 
different  scales  must  be  used.  This, 
of  course,  is  objectionable  and,  hence, 
recourse  is  had  to  a  trimetric  draw- 
ing. A  trimetric  drawing,  therefore, 
requires  three  axes.  The  angles  be- 
tween the  axes  differ,  but  no  one 

angle  can  be  a  right  angle  in  any  case.*  The  same  scale  is 
applied  to  all  three.  Also,  the  axes  may  have  any  direction, 

*  This  evidently  makes  it  an  "oblique  projection.  It  is  impossible  to 
project  a  cube  orthographically  to  produce  this;  since,  if  two  axes  are  parallel 
to  the  plane  of  projection,  the  third  axis  must  be  perpendicular  and  is  hence 
projected  as  a  point.  The  oblique  projection  of  a  cube,  which  is  turned  as 
it  would  be  in  isometric  drawing,  presents  no  new  feature  since  it  results, 
generally  speaking,  in  a  trimetric  projection. 


FIG.  47. 


56  PARALLEL  PROJECTING-LINE  DRAWING 

so  long  as  attention  is  paid  to  the  angular  relation  be- 
tween them. 

If  isometric  drawings  introduce  the  disturbing  symmetry, 
then  the  trimetric  is  to  be  recommended,  unless  one  of  the  other 
types  of  drawing  is  found  to  be  more  suitable.  No  examples  are 
given  in  this  connection,  because  the  application  of  dimetric  and 
trimetric  drawings  involve  no  new  features.  It  is  only  to  be 
remembered,  that  artistic  taste  may  dictate  the  direction  of  the 
axes,  so  as  to  present  the  best  view  of  the  object  to  be  illustrated. 

411.  Axonometric  projection  and  axonometric  drawing. 

Isometric,  dimetric,  and  trimetric  projection  form  a  group  which 
may  be  conveniently  styled  as  axonometric  projections.  All 
three  are  the  result  of  the  orthographic  projection  of  a  cube, 
so  that  the  three  principal  axes  are  projected  in  a  manner  as 
already  indicated.  Axonometric  projections  are  therefore  a 
special  case  of  orthographic  projection,  but  their  advantages 
are  sufficiently  prominent  to  warrant  separate  classification. 

For  isometric  projection,  one  scale  is  used  throughout;  for 
dimetric  projection,  two  separate  scales  are  used;  and  for  tri- 
metric projection,  three  distinct  scales  are  used.  When  applying 
axonometric  projections  to  drawing,  the  same  scale  is  used  on 
all  axes,  and  the  group  then  represents  a  series  which  may  be 
called  axonometric  drawing. 

The  distinction  between  isometric  projection  and  isometric 
drawing  has  been  pointed  out  (Art.  403).  It  becomes  more 
prominent,  however,  in  dimetric  and  trimetric  projection. 

412.  Commercial  application  of  axonometric  projection. 

For  objects  of  simple  shape,  with  few  curves,  isometric  drawings 
serve  a  useful  purpose,  because  they  are  easily  made  and  are 
easily  read  by  those  unfamiliar  with  drawing  in  general.  When 
curves  are  frequent  and  it  is  desirable  to  picture  objects  in  a 
single  view,  oblique  projections  offer  advantages  over  isometric 
drawings  because  it  may  be  possible  to  make  the  planes  of  the 
curves  parallel  to  the  plane  of  projection.  The  curves  will  then 
be  projected  as  they  actually  appear.  When  curves  appear 
in  many  planes,  then  orthographic  projections  will  answer  require- 
ments best,  but,  as  already  mentioned,  their  reading  is  more 
difficult,  due  to  the  simultaneous  interpretation  of  two  or  more 
views. 


AXONOMETRIC  PROJECTION  57 

Before  dismissal  of  the  subject  of  projections  in  general, 
attention  is  called  to  Fig.  48  which  is  a  peculiar  application 
of  isometric  drawing.  It  is  frequently  used  as  an  example  of 
an  optical  illusion.  By  concentrating 
the  vision  at  the  centre  of  the  picture, 
there  seems  to  be  a  sudden  change  from 
six  to  seven  cubes,  or  vice  versa,  depend- 
ing upon  whether  the  central  corner  be 
regarded  as  a  projecting  or  a  depressed 
corner.  This  is  due  to  the  fact  that  all 
of  the  cubes  are  shown  of  the  same  size, 
a  condition  which  is  contrary  to  our 
everyday  experience.  As  objects  recede 
from  the  eye,  they  appear  smaller;  and 

in  isometric  projection  there  is  no  correction  for  this.  In  fact,  all 
the  projections  so  far  considered,  draw  on  the  imagination  for 
their  interpretation,*  and,  therefore,  they  cannot  present  a  per- 
fectly natural  appearance. 

413.  Classification  of  projections.  All  projections  having 
parallel  projecting  lines  may  be  classified  according  to  the 
method  by  which  they  are  made.  This  classification  furnishes 
a  useful  survey  of  the  entire  subject  and  also  serves  to  emphasize 
the  distinction  between  the  different  methods. 

It  will  be  found,  on  analysis,  that  if  the  object  be  conceived 
in  space  with  its  parallel  projecting  lines,  the  oblique  projections 
result  when  the  plane  of  projection  cuts  the  projecting  lines 
obliquely.  When  the  plane  of  projection  cuts  the  projecting 
lines  at  a  right  angle,  then  the  orthographic  series  of  projections 
arise.  If,  still  further,  the  projecting  lines,  coincide  with  some 
of  the  principal  lines  on  the  object,  and  the  plane  of  projection 
is  at  right  angles  to  the  projecting  lines,  then  the  two-dimension 
orthographic  projections  result,  and  these  are  commonly  called 
mechanical  drawings.  If  the  projecting  lines  do  not  coincide 
with  some  of  the  principal  lines  on  the  object,  then  the  axono- 
metric  series  of  projections  follow. 

"The  projections  that  overcome  these  objections  are  known  as  Per- 
spective Projections.  In  these,  the  projecting  lines  converge  to  a  point  at 
which  the  observer  is  supposed  to  be  located.  Photographs  are  perspectives 
in  a  broad  sense. 


58 


PARALLEL  PROJECTING-LINE  DRAWING 


CLASSIFICATION  OF  PROJECTIONS  HAVING  PARALLEL 
PROJECTING  LINES 


All  lines  projected 
equal  in  length. 
Hence,  same  scale 
used  on  all. 

Lines  parallel  to 
plane  of  projection 
drawn  to  equal 
length.  Lines  per- 
pendicular  to 
plane,  projected  as 
shorter  lines  to  di- 
minish distortion. 

Principal  lines  of 
object  parallel  or 
perpendicular  to 
planes  of  projec- 
tion. Sometimes 
called  O  r  t  h  o  - 
graphic  projec- 
tion. 

Isometric.  All 
three  axes  of  cube 
projected  equally, 
hence,  axes  are 
120°  apart  and 
equal  in  length. 
Applied  as  Iso- 
metric Drawing. 

Dimetric.  Two  of 
the  three  axes 
projected  equally, 
hence,  only  two 
angles  equal.  Ap- 
plied as  Dimetric 
Drawing. 

Trimetric.  Axes 
projected  unequ- 
ally, hence,  all 
three  angles  differ. 
Applied  as  Tri- 
metric Drawing. 


Inclination  of 

projecting 

lines    45° 

Oblique.  Pro- 

with plane  of 

• 

jecting  lines 

projection. 

inclined      to 

plane  of  pro- 

Inclination of 

jection     but 

projecting 

parallel      to 

lines  greater 

each  other. 

than     45° 

with  plane  of 

projection. 

Mechanical 

'  On  Plane 

Drawing. 

Surfaces: 

showing  two 

principal  di- 

mensions on  . 

a  single  view. 

Hence,  at 

least    two 

views  needed 

Orthographic. 

'• 

Projecting 

lines  perpen- 

dicularto 

the  plane  of 

projection. 

PROJEC- 
TIONS 

Axonometric. 

projection, 

showing 

three  dimen- 

sions    in     a 

single  view. 

On        r 

Curved    J  Not  used  in  Engineering  drawing. 
Surfaces: 

AXONOMETRIC  PROJECTION  59 


QUESTIONS  OX   CHAPTER  IV 

1.  What  are  the  isometric  axes? 

2.  How  are  they  obtained? 

3.  What  is  the  angular  relation  between  the  pairs  of  axes? 

4.  Show  why  the  isometric  projection  is  a  special  case  of  orthographic. 

5.  What  is  the  distinction  between  isometric  projection  and  isometric 

drawing? 

6.  What  direction  do  the  axes  have  for  their  convenient  application 

to  drawing? 

7.  How  is  a  circle  projected  isometrically? 

8.  Show  the  approximate  method  of  drawing  the  isometric  circle. 

9.  How  are  inclined  lines  laid  off  isometrically? 

10.  How  are  angles  laid  off  isometrically? 

11.  Show  that  the  laying  off  of  an  inclined  line  is  accomplished  by  laying 

off  the  components  of  the  line. 

12.  How  is  the  isometric  circle  graduated  in  the  top  face  of  the  cube? 

13.  Show  how  the  graduation  is  accomplished  in  one  of  the  side  faces 

of  the  cube. 

14.  What  is  a  dimetric  projection? 

15.  What  is  a  dimetric  drawing? 

16.  What  angular  relation  exists  between  the  axes  of  a  dimetric  pro- 

jection? 

17.  What  is  a  trimetric  projection? 

18.  What  is  a  trimetric  drawing? 

19.  What  angular  relation  exists  between  the  axes  of  a  trimetric  drawing? 

20.  Show  why  the  trimetric  drawing  eliminates  the  disturbing  symmetry 

of  an  isometric  drawing. 

21.  Why  cannot  the  angle  between  one  pair  of  axes  be  a  right  angle? 

22.  What  are  axonometric  projections? 

23.  Why  do  projections  with  parallel  projecting  lines  draw  on  the  imag- 

ination for  their  interpretation? 

24.  Draw  the  object  of  Question  33  in  Chapter  2  in  isometric  drawing. 

Use  bounding  figure. 

25.  Draw  a  triangular  prism  in  isometric  drawing. .    Use  a  bounding 

figure. 

26.  Make  an  isometric  drawing  of  a  hexagonal  prism.     Use  a  bounding 

figure. 

27.  Make  an  isometric  drawing  of  a  hexagonal  pyramid.     Use  a  bounding 

figure. 

28.  Make  an  isometric  drawing  of  3-A  (Question  in  Chapter  3). 

29.  Make  an  isometric  drawing  of  3-B. 

30.  Make  an  isometric  drawing  of  3-C. 

31.  Make  an  isometric  drawing  of  3-D. 

32.  Make  an  isometric  drawing  of  3-F. 

33.  Make  an  isometric  drawing  of  3-G. 

34.  Make  an  isometric  drawing  of  3-H. 


60.  PARALLEL  PROJECTING-LINE  DRAWING 

35.  Make  an  isometric  drawing  of  3-1. 

36.  Make  an  isometric  drawing  of  3-J. 

37.  Make  an  isometric  drawing  of  3-K. 

38.  Make  an  isometric  drawing  of  3-L. 

39.  Make  a  complete  classification  of  all  projections  having  parallel 

projecting  lines. 


PART  II 

GEOMETRICAL   PROBLEMS   IN    ORTHOGRAPHIC 
PROJECTION 


CHAPTER  V 
REPRESENTATION  OF  LINES  AND  POINTS 

501.  Introductory.  Material  objects  are  bounded  by  sur- 
faces, which  may  be  plane  or  curved  in  any  conceivable  way. 
The  surfaces  themselves  are  limited  by  lines,  the  forms  of  which 
may  be  straight  or  curved.  Still  further,  these  lines  terminate 
in  points.  Thus,  a  solid  is  really  made  up  of  surfaces,  lines  and 
points,  in  their  infinite  number  of  combinations;  and  these  may 
be  considered  as  the  mathematical  elements  that  make  up  the 
solid.  The  mathematical  elements  must  be  considered  as  concepts 
as  they  have  no  material  existence  and,  hence,  are  purely  imagin- 
ative; that  is,  a  surface  has  no  thickness,  therefore,  it  has  no 
volume.  A  line  or  a  point  is  a  still  further  reduction  along  this 
line  of  reasoning.  The  usefulness  of  these  concepts  must  be 
admitted,  however,  in  view  of  the  fact  that  they  play  such  an 
important  role  in  the  conception  of  objects. 

In  general,  the  outline  of  any  object  is  found  by  intuitively 
locating  certain  points,  and  joining  the  points  by  proper-  lines; 
the  lines,  when  taken  in  their  proper  order  determine  certain 
surfaces,  and  the  space  included  between  them  forms  the  solid 
(the  material  object)  in  question.  Hence,  to  view  material  objects 
analytically,  the  nature  of  their  mathematical  elements  must  be 
known. 

In  Part  II  of  this  book,  the  graphical  representation  of  mathe- 
matical concepts  engages  the  attention.  Whether  the  treatment 
of  the  subject  be  from  the  viewpoint  of  mathematics  or  of  drawing, 
entirely  depends  upon  the  ultimate  use.  In  the  two  chapters  to 

61 


62  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

be  presented  (V  and  VI),  the  method  of  representing  lines, 
points  and  planes,  orthographically,  will  be  discussed.  So  far, 
attention  has  been  directed  to  the  representation  of  material 
objects,  as  common  every  day  experience  renders  such  objects 
familiar  to  us. 

Before  proceeding  with  the  subject  in  all  its  detail,  familiarity 
must  be  gained  with  certain  fundamental  operations  on  lines, 
points  and  planes,  as  well  as  with  their  graphical  representation 
on  two  assumed  planes  called  the  planes  of  projection.  The 
fundamental  operations  are  grouped  and  studied  without  apparent 
reference  to  future  application.  It  is  desirable  to  do  this  so  as 
to  avoid  frequent  interruptions  in  the  chain  of  reasoning  when 
applying  the  operations  to  the  solution  of  problems. 

The  student  will  save  considerable  time  if  he  is  well  versed 
in  the  fundamental  operations.  In  view  of  this,  many  questions 
are  given  at  the  end  of  the  chapters  so  that  his  grasp  of  the  sub- 
ject may  be  tested  from  time  to  time.  It  may  be  needless  to  say 
that  the  solution  of  subsequent  problems  is  utterly  impossible 
without  this  thorough  grounding.  Frequent  sketches  should 
be  made,  representing  lines,  points  and  planes  in  positions  other 
than  shown  in  this  book.  These  sketches  should  be  made  both 
in  orthographic  and  in  oblique  projection.  By  this  means,  the 
student  will  increase  his  experience  in  the  subject,  much  more 
than  is  possible  by  all  the  reading  he  might  do.  The  proof  of 
one's  ability  always  lies  in  the  correct  execution  of  the  ideas 
presented.  The  subject  under  consideration  is  a  graphical  one, 
and,  as  such,  drawing  forms  the  test  mentioned.  It  ha§  been 
considered  necessary  to  caution  the  student  so  as  to  avoid  the 
complications  that  will  result  later,  due  to  insufficient  preparation. 
Only  in  this  way  will  the  subject  become  of  interest,  to  say  nothing 
of  its  importance  in  subsequent  commercial  applications. 

502.  Representation  of  the  line.  Let  Fig.  49  be  an  oblique 
projection  of  two  planes  intersecting  at  right  angles  to  each 
other.  The  plane  W  is  called  the  vertical  plane  of  projection 
and  HH  is  called  the  horizontal  plane  of  projection;  these  planes 
intersect  in  a  line  XY  which  is  termed  the  ground  line.  The 
two  planes  taken  as  a 'whole  are  known  as  the  principal  planes; 
and  by  their  intersection,  they  form  four  angles,  numbered  as 
shown  in  the  figure.  In  what  immediately  follows,  attention 


REPRESENTATION  OF  LINES  AND  POINTS 


63 


will  be  concentrated  on  operations  in  the  first  angle  of  projection, 
and  later,  extended  to  all  four  angles. 

Suppose  a  cube  ABCDEFGH  is  located  in  the  first  angle  of 
projection,  so  that  one  face  lies  wholly  in  the  horizontal  plane 
and  another  face  lies  wholly  in  the  vertical  plane.  The  two  faces 
then  lying  in  the  principal  planes,  intersect  in  a  line  which  coin- 
cides with  the  ground  line  for  the  conditions  assumed.  Suppose, 
further,  that  it  is  ultimately  desired  orthographically  to  represent 
the  diagonal  AG  of  the  cube.  For  the  present,  the  reasoning 
will  be  carried  out  by  the  aid  of  the  oblique  projection. 

The  construction  of  the  horizontal  projection  consists  in  drop- 
ping upon  the  horizontal  plane,  perpendiculars  from  points  on 


FIG.  49. 


the  line.  For  the  line  AG,  the  projecting  perpendicular  from 
the  point  A  is  AE,  and  E  is  then  the  horizontal  projection  of  the 
point  A  in  space.*  As  for  the  point  G  on  the  line,  that  already 
lies  in  the  horizontal  plane  and  is  its  own  horizontal  projection. 
Thus,  two  horizontal  projections  of  two  points  on  the  line  are 
established,  and,  hence,  the  horizontal  projection  of  the  line  is 
determined  by  joining  these  two  projections.  This  is  true, 
because  all  the  perpendiculars  from  the  various  points  on  the  line 
lie  in  a  plane,  which  is  virtually  the  horizontal  projecting  plane 
of  the  line.  It  cuts  the  horizontal  plane  of  projection  in  a  line 
EG,  which  is  the  horizontal  projection  of  the  line  AG  in  space. 

*  It  is  to  be  noted  that  the  projection  of  a  point  is  found  at  the  place 
where  the  projecting  line  pierces  the  plane  of  projection. 


64 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


Putting  this  in  another  form,  the  projection  EG  gives  the  same 
mental  impression  to  an  abserver  viewing  the  horizontal  plane 
from  above  as  does  the  line  AG  itself;  in  fact,  EG  is  a  drawing 
of  the  line  AG  in  space. 

Directing  attention  for  a  moment  to  the  vertical  plane,  it  is 
found  that  the  construction  of  the  vertical  projection  consists 
in  dropping  a  series  of  perpendiculars  from  the  line  AG  to  that 
plane.  For  point  G,  the  perpendicular  to  the  vertical  plane  is 
GH,  and  H  is  thus  the  vertical  projection  of  G.  The  point  A 
lies  in  the  vertical  plane  and,  hence,  is  its  own  vertical  projection. 
A  line  joining  A  and  H  is  the  vertical  projection  of  AG.  Again, 
a  plane  passed  through  AG,  perpendicular  to  the  vertical  plane, 


FIG.  49. 


will  cut  from  it  the  line  AH,  which  is  the  vertical  projection  as 
has  been  determined.  Thus,  AH  is  a  drawing  of  AG  because  it 
conveys  the  same  mental  impression  to  an  observer  who  views 
it  in  the  way  the  projection  was  made. 

503.  Line  fixed  in  space  by  its  projections.  The  location 
of  the  principal  planes  is  entirely  arbitrary,  as  is  also  the  line  in 
Question;  but,  when  both  are  once  assumed,  the  line  is  fixed  in 
space  by  its  projections  on  the  principal  planes.  If  a  plane  be 
passed  through  the  horizontal  projection  EG  (Fig.  49),  perpen- 
dicular to  the  horizontal  plane,  it  will  contain  the  line  AG  in 
space,  since  the  method  is  just  the  reverse  of  that  employed  in 
finding  the  projection.  Similarly,  a  plane  through  the  vertical 
projection,  perpendicular  to  the  vertical  plane,  will  also  contain 


REPRESENTATION  OF  LINES  AND  POINTS 


65 


the  line  AG.  It  naturally  follows  that  the  line  AG  is  the  inter- 
section of  the  horizontal  and  vertical  projecting  planes  and, 
therefore,  the  line  is  absolutely  fixed,  with  reference  to  the  prin- 
cipal planes,  by  its  projections  on  those  planes. 

504.  Orthographic  representation  of  a  line.  The  line  that 
has  been  considered  so  far  is  again  represented  in  the  left-hand 
view  of  Fig.  50,  in  oblique  projection.  The  edges  of  the  cube 
have  been  omitted  here  in  order  to  concentrate  attention  to  the 
line  in  question.  AG  is  that  line,  as  before,  EG  is  its  horizontal 
projection  and  AH  is  its  vertical  projection.* 

Suppose  that  the  plane  HH  is  revolved  in  the  direction  of  the 
arrows,  90°  from  its  present  position,  until  it  coincides  with  the 


FIG.  50. 


vertical  plane  W.  The  view  on  the  right  of  Fig.  50  shows  the 
resultant  state  of  affairs  in  orthographic  projection.  AH  is  the 
vertical  projection  and  EG  is  the  horizontal  projection.  AE  in 
one  view  is  the  equivalent  of  AE  in  the  other;  HG  and  EH  in 
one  view  are  the  equivalents  of  HG  and  EH  in  the  other.  Both 
views  represent  the  same  line  AG  in  space.  At  first  sight,  it  may 
appear  that  the  oblique  projection  is  sufficiently  clear,  and  such 
is  the  case;  but,  in  the  solution  of  problems,  the  orthographic 
projection  as  shown  on  the  right  possesses  many  advantages.  In 
due  time,  this  mode  of  representation  will  be  considered,  alone, 
without  the  oblique  projection. 

*  The  projections  of  a  line  may  be  considered  as  shadows  on  their  respective 
planes.  The  light  comes  in  parallel  rays,  perpendicularly  directed  to  the 
planes  of  projection.  See  also  Arts.  203  and  310  in  this  connection. 


66 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


505.  Transfer  of  diagrams  from  orthographic  to  oblique 
projection.  It  is  desirable  to  know  how  to  transfer  diagrams 
from  one  kind  of  projection  to  the  other.  If  the  orthographic 
projection  appears  confusing,  the  transfer  to  oblique  projection 
may  be  of  service.  On  the  right  of  Fig.  50,  is  given  the  ortho- 
graphic projection  of  the  line  AG.  To  construct  the  oblique 
projection  from  this,  draw  the  principal  planes  as  shown  in  the 
left-hand  diagram.  From  any  point  E  on  the  ground  line  in  the 
oblique  projection,  lay  off  EA,  vertically,  equal  to  EA  in  the 
orthographic  projection.  On  the  sloping  line  (ground  line), 
lay  off  EH  equal  to  EH  in  the  orthographic  projection.  Then 
draw  HG  in  the  horizontal  plane,  equal  to  HG  in  the  orthographic 
projection.  A  line  joining  A  and  G  in  the  oblique  projection 


FIG.  50. 


gives  the  actual  line  in  space;  hence,  in  the  oblique  projection, 
the  actual  line  and  both  of  its  projections  are  shown.  In  the 
orthographic  projection,  only  the  projections  are  given;  the 
line  itself  must  be  imagined.  Compare  the  method  of  constructing 
the  oblique  projection  with  Art.  207  and  note  the  similarity. 

506.  Piercing  points  of    lines  on  the  principal  planes. 

If  AG  (Fig.  50)  be  considered  as  a  limited  portion  of  line  indefi- 
nitely extended  in  both  directions,  then  A  and  G  are  the  piercing 
points  on  the  vertical  and  horizontal  planes  respectively.  On 
observation,  it  will  be  found  that  the  vertical  piercing  point  lies 
on  the  vertical  projection  of  the  line,  and  also  on  a  perpendicular 
to  the  ground  line  XY  from  the  point  where  the  horizontal  pro- 
jection intersects  the  ground  line;  hence,  it  is  at  their  intersection. 


REPRESENTATION   OF  LINES  AND  POINTS 


67 


Likewise,  the  horizontal  piercing  point  lies  on  the  horizontal 
projection  of  the  line  and  also  on  the  perpendicular  erected  at 
the  intersection  of  the  vertical  projection  with  the  ground  line; 
hence,  again,  it  is  at  the  intersection  of  these  two  lines. 

In  Fig.  51  let  ab  be  the  horizontal  projection  of  a  line  AB  in 
space,  and  a'b'  be  the  corresponding  vertical  projection.  The 
line,  if  extended,  would  pierce  the  horizontal  plane  at  c  and  the 
vertical  plane  at  d'.  This  line  is  shown  as  an  oblique  projection 
in  Fig.  52.  All  the  lines  that  are  required  in  the  mental  process 
are  shown  in  this  latter  view.  The  actual  construction  in  ortho- 
graphic projection  is  given  in  Fig.  51.  To  locate  the  vertical 
piercing  point  prolong  both  projections,  and  at  the  point  where 
the  horizontal  projection  intersects  the  ground  line,  erect  a  per- 


FIG.  51. 


FIG.  52. 


pendicular  until  it  intersects  the  prolongation  of  the  vertical 
projection.  This  locates  the  vertical  piercing  point.  To  find 
the  horizontal  piercing  point,  prolong  the  vertical  projection 
until  it  intersects  the  ground  line,  and  at  this  point,  erect  a  per- 
pendicular to  the  ground  line.  Then,  the  point  at  which  this 
perpendicular  intersects  the  prolongation  of  the  horizontal  pro- 
pection  is  the  horizontal  piercing  point. 

A  convenient  way  of  looking  at  cases  of  this  kind  is  to  assume 
that  XY  is  the  edge  of  the  horizontal  plane  while  viewing  the  verti- 
cal projection,  therefore,  AB  must  pierce  the  horizontal  plane 
somewhere  in  a  line  perpendicular  to  the  vertical  plane  at  the 
point  c'.  In  viewing  the  horizontal  plane,  XY  now  represents 
the  vertical  plane  on  edge.  Here,  again,  the  line  AB  must  pierce 


68 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


the  vertical  plane  somewhere  in  a  line  perpendicular  to  the  hori- 
zontal plane  at  the  point  d. 

507.  Nomenclature  of  projections.     In  what  follows,  the 
actual  object  will  be  designated  by  the  capital  letters,  as  the  line 
AB  for  instance.     The  horizontal  projections  will  be  indicated 
by  the  small  letters  as  the  line  ab,  and  the  vertical  projections 
by  the  small  prime  letters  as  the  line  a'b'.     It  should  always  be 
remembered   that  in   orthographic   projection,   the   projections 
alone  are  given,  the  actual  object  is  to  be  imagined. 

508.  Representation  of  points.     A  single  point  in  space  is 
located  with  respect  to  the  principal  planes  as  shown  in  Fig.  53. 
A  is  the  actual  point  while  a  is  its  horizontal,  and  a'  its  vertical 
projection.     The  distance  of  A  above  the  horizontal  plane  is  equal 
to  the  length  of  its  projecting  perpendicular  Aa  and  this  is  equal 
to  a'o  because  Aa  and  a'o  are  both   perpendicular  to  the  hori- 


FIG.  53. 


zontal  plane  HH.  Also,  Aa'  and  ao  are  perpendicular  to  the  verti- 
cal plane,  therefore,  the  figure  Aaoa'  is  a  rectangle,  whose  opposite 
sides  are  necessarily  equal.  Hence,  also,  the  distance  of  A  from 
the  vertical  plane  is  equal  to  the  length  of  its  projecting  perpen- 
dicular Aa'  which  also  equals  ao.  Performing  the  usual  revolu- 
tion of  the  horizontal  plane,  a  will  reach  a"  on  a  line  a'a"  which 
is  perpendicular  to  the  ground  line  XY.  It  has  been  shown  that 
a'o  is  perpendicular  to  XY  and  it  only  remains  to  prove  that  oa" 
is  a  continuation  of  a'o.  This  must  be  so,  because  a  revolves 
about  the  ground  line  as  an  axis,  in  a  plane  determined  by  the 
two  intersecting  lines  a'o  and  ao.  This  plane  cuts  from  the  ver- 


REPRESENTATION  OF  LINES  AND  POINTS  69 

tical  plane  the  line  a'a"  which  is  perpendicular  to  XY  because  a 
portion  (a'o)  of  it  is  perpendicular.  As  the  point  a  revolves 
about  XY  as  an  axis,  it  describes  a  circle,  whose  radius  is  oa, 
and  hence  oa"  must  equal  oa. 

In  projection,*  this  is  shown  on  the  right-hand  diagram  of 
Fig.  53.  Both  figures  are  lettered  to  correspond  as  far  as  con- 
sistent. The  actual  point  A  is  omitted,  however,  in  the  right- 
hand  diagram  because  the  very  object  of  this  scheme  of  repre- 
sentation is  to  locate  the  point  from  two  arbitrary  planes  (prin- 
cipal planes),  solely  by  their  projections  on  those  planes.  This 
latter  is  an  exceedingly  important  fact  and  should  always  be 
borne  in  mind. 

509.  Points   lying  in  the    principal  planes.     If  a  point 
lies  in  one  of  the  principal  planes,  it  is  its  own  projection  in  that 
plane  and  its  corresponding    projection 

lies  in  the  ground  line.     Fig.  54  shows 
such  cases  in  projection.    A  is  a  point  \a         $      Cc> 

lying  in  the  vertical  plane,  at  a  distance 
a'a  above  the  horizontal  plane;   its  ver-  [b 

tical  projection  is  a'  and  its  correspond-  -pio.  54. 

ing    horizontal    projection    is   a.     B    is 

another  point,  lying  in  the  horizontal  plane,  at  a  distance  bb' 
from  the  vertical  plane;  b  is  its  horizontal  and  b'  the  correspond- 
ing vertical  projection. 

If  a  point  lies  in  both  planes,  the  point  coincides  with  both 
of  its  projections  and  must  therefore  be  in  the  ground  line.  C  is 
such  a  point,  and  its  two  projections  are  indicated  by  cc',  both 
letters  being  affixed  to  the  one  point.  The  use  of  these  cases 
will  appear  as  the  subject  develops. 

510.  Mechanical  representation  of  the  principal  planes. 

For  the  time  being,  the  reader  may  find  it  desirable  to  construct 
two  planes  f  so  that  lines  and  points  may  be  actually  represented 

*  Hereafter,  orthographic  projection  will  usually  be  designated  simply 
as  "in  projection." 

t  For  classroom  work,  a  more  serviceable  device  can  be  made  of  hinged 
screens,  constructed  of  a  fine  mesh  wire.  Wires  can  be  easily  inserted  to 
represent  lines  and  the  projections  drawn  with  chalk.  The  revolution  of 
the  planes  can  be  accomplished  by  properly  hinging  the  planes  so  that  they 
can  be  made  to  lie  approximately  flat. 


70 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


FIG.  55. 


with  reference  to  them.     If  two  cards  be  slit  as  shown  in  Fig.  55, 
they  can  be  put  together  so  as  to  represent  two  planes  at  right 

angles  to  eaj&h  other.  Lines  may  be 
represented  by  use  of  match-sticks  and 
points  by  pin-heads,  the  pin  being  so 
inserted  as  to  represent  its  projecting 
perpendicular  to  one  plane.  The  idea 
is  recommended  until  the  student  be- 
comes familiar  with  the  involved  opera- 
tions. As  soon  as  this  familiarity  is  obtained,  the  cards  should 
be  dispensed  with,  and  the  operations  reasoned  out  [in  space,  as 
far  as  possible,  without  the  use  of  any  diagrams. 


611.  Lines  parallel  to  the  planes  of  projection.  Assume 
a  Ime  parallel  to  the  horizontal  plane.  Evidently  this  line  cannot 
pierce  the  horizontal  plane  on  account  of  the  parallelism.  It 
will  pierce  the  vertical  plane,  however,  if  it  is  inclined  to  that  plane . 


FIG.  56. 


In  Fig.  56  this  line  is  shown  both  in  oblique  and  orthographic 
projection.  The  piercing  point  on  the  vertical  plane  is  found 
by  erecting  a  perpendicular  at  the  point  where  the  horizontal 
projection  intersects  the  ground  line.  It  will  also  lie  on  the  ver- 
tical projection  of  the  line.  As  the  line  is  parallel  to  the  horizontal 
plane,  its  vertical  projection  is  parallel  to  the  ground  line.  The 
piercing  point  will  therefore  lie  at  the  intersection  of  the  two 
lines;  that  is,  of  the  perpendicular  from  the  ground  line  and  the 
vertical  projection  of  the  line.  In  both  views  of  the  figure,  AB 
is  the  given  line  and  b'  is  its  piercing  point. 

If,  on  the  other  hand,  the  line  is  parallel  to  the  vertical  plane 


REPRESENTATION  OF  LINES  AND   POINTS 


71 


(Fig.  57)  and  inclined  to  the  horizontal  plane,  it  will  pierce  the 
horizontal  plane  at  some  one  point.  Its  horizontal  projection 
is  now  parallel  to  the  ground  line  XY.  The  horizontal  piercing 


FIG.  57. 

point  is  at  b,  as  shown,  and  is  found  in  much  the  same  way  as  that 
in  the  illustration  immediately  preceding. 

A  case  where  the  line  is  parallel  to  both  planes  is  shown  in 


FIG.  58. 

Fig.  58.     This  line  cannot  pierce  either  plane  and,  therefore,  both 
its  projections  must  be  parallel  to  the  ground  line  as  depicted. 

512.  Lines  lying  in  the  planes  of  projection.  If  a  line 
lies  in  the  plane  of  projection,  it  is 
its  own  projection  in  that  plane,  and 
its  corresponding  projection  lies  in 
the  ground  line.  When  the  line  lies 
in  both  planes  of  projection  it  must 
therefore  coincide  with  the  ground  line. 
Fig.  59  shows  in  projection  the  three 
cases  possible.  The  first  is  a  line 
lying  in  the  horizontal  plane,  ab  is  its  horizontal  projection  and 


72  GEOMETEICAL  PROBLEMS  IN   PROJECTION 

a'b'  is  its  corresponding  vertical  projection.  The  second  is  a 

line  lying  in  the  vertical  plane,  c'd' 
is  its  vertical  projection  and  cd  is  its 
corresponding  horizontal  projection. 
Y  The  third  case  shows  a  line  in  both 
planes  and  its  horizontal  and  vertical 
projections  coincide  with  the  ground 
FIG  59.  line  and  also  the  line  itself;  the 

coincident  projections  are  indicated 

as  shown  at  ee'  and  ff,  read  ef  and  e'f. 

513.  Lines  perpendicular  to  the  planes  of  projection. 

If  a  line  is  perpendicular  to  the  horizontal  plane,  its  projection 
on  that  plane  is  a  point,  because  both  projecting  perpendiculars 
from  the  extremities  must  coincide  with  the  perpendicular  line. 
The  vertical  projection,  however,  shows  the  line  in  its  true  length, 
perpendicular  to  the  ground  line,  as  such  a  line  is  'parallel  to  the 


f      '"'     tf 
TT 

"Vl                      *         V     rr            -1 

V 

c 

2          ^« 

X                     c     1    n        4 

3 

Ut 

,1 

V 

FIG.  60. 

vertical  plane.  Fig.  60  shows  this  in  projection  as  AB,  and,  in 
addition  a  profile  plane  is  added  which  indicates  the  fact  more 
clearly.  CD  is  a  line  perpendicular  to  the  vertical  plane  with  one 
extremity  of  the  line  in  that  plane.  In  the  profile  plane,  the 
lines  AB  and  CD  appear  to  intersect,  but  this  is  not  necessarily 
the  case.  The  construction  of  the  projections  of  the  line  CD  is 
identical  with  that  immediately  preceding. 

514.  Lines  in  all  angles.  So  far,  the  discussion  of  lines 
and  points  in  space  has  been  limited  entirely  to  the  first  angle. 
If  a  line  is  indefinite  in  extent,  it  may  pass  through  several  of  the 
four  angles.  A  case  in  each  angle  will  be  taken  and  the  salient 
features  of  its  projections  will  be  pointed  out. 

Fig.  61  shows  a  line  passing  through  the  first  angle.  It  con- 
tinues beyond  into  the  second  and  fourth  angles.  In  projection, 


REPRESENTATION  OF  LINES  AND  POINTS 


73 


the  condition  is  shown  on  the  right.     The  projections  ab  and  a'b' 
show  those  of  the  limited  position  that  traverses  the  first  angle. 


The  horizontal 


The  dotted  extensions  show  the  projection  of  the  indefinite  line, 
and  are  continued,  at  pleasure,  to  any  extent. 

A  line  in  the  second  angle  is  shown  in  Fig.  62. 
projection  is  ab  and  the  ver- 
tical projection  is  a'b'.  If 
the  horizontal  plane  be  re- 
volved into  coincidence  with 
the  vertical  plane,  the  view 
in  projection  will  show  that, 
in  this  particular  case,  the 
projections  of  the  lines  cross  pIG  62 

each  other.    Only  the  limited 

portion  in  the  second  angle  is  shown,  although  the  line  may  be 
indefinitely  extended  in  both  directions. 

A  third  angle  line  is  illustrated  in  Fig.  63.     It  may  be  observed, 


FIG.  63. 

in  comparison  with  a  line  in  the  first  angle,  that  the  horizontal 
and  vertical  projections  are  interchanged  for  the  limited  portion 


74 


GEOMETRICAL  PROBLEMS  IN    PROJECTION 


of  the  line  shown.  In  other  words  the  horizontal  projection 
of  the  line  is  above  the  ground  line  and  the  vertical  projection 
is  below  it.  The  case  may  be  contrasted  with  Fig.  61. 

The  last  case  is  shown  in  the  fourth  angle  and  Fig.  64  depicts 


FIG.  64. 

this  condition.  Both  projections  now  cross  each  other  below 
the  ground  line.  Contrast  this  with  Fig.  62.  The  similarity 
of  the  foregoing  and  Arts.  315,  316  and  317  may  be  noted. 

515.  Lines  with  coincident  projections.  Lines,  both  of 
whose  projections  lie  in  the  ground  line  have  been  previously 
considered  (511).  If  a  line  passes  through  the  ground  line  from 


66' 


FIG.  65. 

the  second  to  the  fourth  angle,  so  that  any  point  on  the  line  is 
equidistant  from  both  planes,  then  the  two  projections  of  the 
line  will  be  coincident.  Fig.  65  illustrates  such  a  condition. 
The  line  is  not  indeterminate  by  having  coincident  projections, 
however,  because  the  oblique  projections  can  be  constructed 


REPRESENTATION  OF  LINES  AND  POINTS  75 

from  the  orthographic  representation.  The  projection  on  the 
profile  plane  shows  that  this  line  bisects  the  second  and  fourth 
angles. 

If  the  line  passes  through  the  ground  line,  from  the  second 
to  the  fourth  angles,  but  is  not  equidistant  from  the  principal 
planes,  then  both  projections  will  pass  through  the  same  point 
on  the  ground  line. 

516.  Points  in  all  angles.  Fig.  66  shows  four  points  A,  B,  C, 
and  D,  lying,  one  in  each  angle.  The  necessary  construction 
lines  are  shown.  On  the  right,  the  condition  is  depicted  in 
projection,  the  number  close  to  the  projection  indicating  the  angle 
in  which  the  point  is  located. 

Observation  will  show  that  the  first  and  third  angle  projec- 


t   !' 


H  2  |3 

I  ! 


-'      fr 


FIG.  66. 


tions  are  similar  in  general  appearance,  but  with  projections 
interchanged.  The  same  is  true  of  the  second  and  fourth  angles; 
although,  in  the  latter  cases,  both  projections  fall  to  one  or  the 
other  side  of  the  ground  line. 

517.  Points   with     coincident     projections.     It  may  be 

further  observed  in  Fig.  66  that  in  the  second  and  fourth  angles, 
the  projection  b,  b'  and  d,  d'  may  be  coincident.  This  simply 
means  that  the  points  are  equidistant  from  the  planes  of  projection. 
The  case  of  the  point  in  the  ground  line  has  been  noted 
(508),  such  points  lie  in  no  particular  angle,  unless  a  new  set  of 
principal  planes  be  introduced.  It  can  then  be  considered  under 
any  case  at  pleasure,  depending  upon  the  location  of  the  prin- 
cipal planes. 

518.  Lines  in  profile  planes.    A  line  may  be  located  so  that 
both  of  its  projections  are  perpendicular  to  the  ground  line. 


76 


GEOMETRICAL  PROBLEMS   IN   PROJECTION 


The  projections  must  therefore  be  coincident,*  since  they  pass 
through  the  same  point  on  the  ground  line.  Fig.  67  shows  an 
example  of  this  kind.  Although  the  actual  line  in  space  can  be 
determined  from  its  horizontal  projection  ab  and  its  vertical 


FIG.  67. 

projection  a'b',  still  this  is  only  true  because  a  limited  portion 
of  the  line  was  chosen  for  the  projections.  The  profile  shown  on 
the  extreme  right  clearly  indicates  the  condition.  The  location 
of  the  profile  with  respect  to  the  projection  should  also  be  noted. 
The  view,  is  that  obtained  by  looking  from  right  to  left,  and  is 
therefore  located  on  the  right  side  of  the  projection.  The  number- 
ing on  the  angles  should  also  help  the  interpretation. 


D 


H 


C 

1« 

b', 

d'l 

X. 

/ 

a' 

1 

2           3 

4 

a 

,,  .. 

„' 

FIG.  GS. 


Fig.  68  shows  a  profile  (on  the  left-hand  diagram)  of  one  line 
in  each  angle.  The  diagram  on  the  right  shows  the  lines  in 
projection.  The  numbers  indicate  the  angle  in  which  the  line 
is  located. 


*  Compare  the  coincident  projections  in  this  case  with  those  of  Art.  515. 


REPRESENTATION  OF  LINES  AND  POINTS  77 


QUESTIONS  ON  CHAPTER  V 

1.  Discuss  the  point,  the  line,  and  the  surface,  and  show  how  the  material 

object  is  made  up  of  them. 

2.  What  are  the  mathematical  elements  of  a  material  object? 

3.  Why  are  the  mathematical  elements  considered  as  concepts? 

4.  How  is  the  outline  of  a  material  object  determined? 

5.  What  is  meant  by  the  graphical  representation  of  mathematical  con- 

cepts? 

6.  What  are  the  principal  planes  of  projection? 

7.  What  is  the  ground  line? 

8.  How  many  dihedral  angles  are  formed  by  the  principal  planes  and 

how  are  they  numbered?     Make  a  diagram. 

9.  How  is  a  line  orthographically  projected  on  the  principal  planes? 

10.  How  many  projections  are  required  to  fix  the  line  with  reference  to 

the  principal  planes?     Why? 

11.  Show  how  one  of  the  principal  planes  is  revolved  so  as  to  represent  a 

line  in  orthographic  projection. 

12.  Assume  a  line  in  the  first  angle  in  orthographic  projection  and  show 

how  the  transfer  is  made  to  an  oblique  projection. 

13.  Under  what  conditions  will  a  line  pierce  a  plane  of  projection? 

14.  Assume  a  line  that  is  inclined  to  both  planes  of  projection  and  show 

how  the  piercing  points  are  determined  orthographically.  Give 
the  reasoning  of  the  operation. 

15.  Do  the  orthographic  projections  of  a  line  represent  the  actual  line  in 

space?    Why? 

16.  Show  a  point  in  oblique  projection  and  also  the  projecting  lines  to  the 

principal  planes.  Draw  the  corresponding  diagram  in  orthographic 
projection  showing  clearly  how  one  of  the  principal  planes  is  re- 
volved. 

17.  Draw,  in  projection,  a  point  lying  in  the  horizontal  plane;  a  point 

lying  in  the  vertical  plane;  a  point  lying  in  both  planes.  Observe 
nomenclature  in  indicating  the  points. 

18.  Indicate  in  what  angle  the  points  shown  d 

in  Fig.  5-A  are  located. 

19.  Draw  a   line   parallel   to  the  horizontal  t<* 

plane  but  inclined  to  the  vertical  plane 
in  orthographic   projection.      Transfer 


diagram  to  oblique  projection.  !  ,   [b> 

20.  Draw  a  line  parallel  to  the  vertical  plane  ;6 

but  inclined  to  the  horizontal  plane  in  'J'J 

orthographic  projection.     Transfer  dia-  FIG.  5-A. 

gram  to  oblique  projection. 

21.  Draw  a  line  parallel  to  both  principal  planes  in  orthographic  pro- 

jection.    Transfer  diagram  to  oblique  projection. 

22.  When  a  line  is  parallel  to  both  principal  planes  is  it  parallel  to  the 

ground  line?    Why? 


78 


GEOMETRICAL  PROBLEMS  IN   PROJECTION 


23.  In  Fig.  5-B,  make  the  oblique  projection  of  the  line  represented. 

24.  Draw  the  projections  of  a  line  lying  in  the  horizontal  plane. 

25.  Draw  the  projections  of  a  line  lying  in  the  vertical  plane. 

26.  Draw  the  projections  of  a  line  lying  in  the  ground  line,  observing  the 

nomenclature  in  the  representation. 


a 

N, 


FIG.  5-B. 


FIG.  5-C. 


27.  In  Fig.  5-C  give  the  location  of  the  lines  represented  by  the  ortho- 

graphic projections.     Construct  the  corresponding  oblique  projec- 
tions. 

28.  Show  two  lines,  one  perpendicular  to  each  of  the  planes  and  also 

draw  a  profile  plane  for  each  indicating  the  advantageous  use  in 
such  cases. 

29.  When  a  line  is  perpendicular  to  the  horizontal  plane,  how  is  it  pro- 

jected on  that  plane? 

30.  When  a  line  is  perpendicular  to  the  vertical  plane,  why  is  the  horizon- 

tal projection  equal  to  it  in  length? 

31.  Draw  a  line  in  the  second  angle,  in  orthographic  projection.     Transfer 

the  diagram  to  oblique  projection. 

32.  Draw  a  line  in  the  third  angle,  in  orthographic  projection.     Transfer 

the  diagram  to  oblique  projection. 

33.  Draw  a  line  in  the  fourth  angle,  in  orthographic  projection.     Transfer 

the  diagram  to  oblique  projection. 

34.  In  Fig.  5-D,  specify  in  which  angles  each  of  the  lines  are  situated. 

g' 


Y           ! 

i     r  Y 

!       i       i     ! 
i—  ~J 

FIG.  5-D. 

35.  Draw  lines  similar  to  5-D  in  orthographic  projection  and  transfer 

the  diagrams  to  oblique  projection. 

36.  Make  the  orthographic  projection  of  a  line  with  coincident  projections 

and  show  by  a  profile  plane  what  it  means.  Take  the  case  where 
the  line  passes  through  one  point  on  the  ground  line  and  is  per- 
pendicular to  it,  and  the  other  case  where  it  is  inclined  to  the 
ground  line  through  one  point. 


REPRESENTATION   OF  LINES  AND  POINTS  79 

37.  Locate  a  point  in  each  angle  and  observe  the  method  of  indicating 

them. 

38.  Make  an  oblique  projection  of  the  points  given  in  Question  37. 

39.  Make  an  orthographic   projection  of  points  with   coincident  pro- 

jections and  show  under  what  conditions  they  become  coincident. 

40.  In  what  angles  are  coincident  projections  of  points  possible?     Show 

by  profile. 

41.  Make  the  oblique  projections  of  the  lines  shown  in  Fig.  68  of  the 

text.    Use  arrows  to  indicate  the  lines. 

42.  Why  is  it  advantageous  to  use  a  profile  plane,  when  the  lines  are 

indefinite  in  extent,  and  lie  in  the  profile  plane? 


CHAPTER  VI 
REPRESENTATION  OF  PLANES 

601.  Traces  of  planes  parallel  to  the  principal  planes. 

Let  Fig.  69  represent  the  two  principal  planes  by  HH  and  W 
intersecting  in  the  ground  line  XY.  Let,  also,  RR  be  another 
plane  passing  through  the  first  and  fourth  angles  and  parallel  to 
W.  The  plane  RR  intersects  the  horizontal  plane  HH  in  a  line 
tr,  which  is  called  the  trace  of  the  plane  RR.  As  RR  is  parallel 


FIG.  69. 

to  W,  the  case  is  that  of  two  parallel  planes  cut  by  a  third  plane, 
and,  from  solid  geometry,  it  is  known  that  their  intersections 
are  parallel.  If  the  horizontal  plane  is  revolved,  by  the  usual 
method,  into  coincidence  with  the  vertical  plane,  the  resulting 
diagram  as  shown  on  the  right  will  be  the  orthographic  represen- 
tation of  the  trace  of  a  plane  which  is  parallel  to  the  vertical 
plane. 

If,  as  in  Fig.  70,  the  plane  is  parallel  to  the  horizontal  plane, 
the  condition  of  two  parallel  planes  cut  by  a  third  plane  again 
presents  itself,  the  vertical  plane  of  projection  now  being  the 
cutting  plane.  The  trace  t'r'  is  then  above  the  ground  line, 
and,  as  before,  parallel  to  it. 

602.  Traces  of  planes  parallel  to  the  ground  line.  When 
a  plane  is  parallel  to  the  ground  line,  and  inclined  to  both  planes 

80 


REPRESENTATION   OF  PLANES 


81 


of  projection,  it  must  intersect  the  principal  planes.     The  line 
of  intersection  on  each  principal  plane  will  be  parallel  to  the  ground 


FIG.  70. 

line  because  the  given  plane  is  parallel  to  the  ground  line  and, 
hence,  cannot  intersect  it.     Fig.  71  shows  this  condition  in  oblique 


R 


FIG.  71. 

and  orthographic  projection,  in  which  tr  is  the  horizontal  trace 
and  t'r'  is  the  vertical  trace. 

A  special  case  of  this  occurs  if  the  plane  passes  through  the 
ground  line.    Both  traces  then  coincide  with 
the  ground  line  and  the  orthographic  repre- 
sentation becomes  indeterminate  unless  the 
profile  plane  is  attached. 

Fig.  72  is  a  profile  and  shows  several 
planes  passing  through  the  ground  line,  each 
of  which  is  now  determined.  It  may  be 
possible,  however,  to  introduce  a  new  hori- 
zontal plane  H'H',  parallel  to  the  principal 
horizontal  plane.  Such  an  artifice  will  bring  the  case  into  that 
immediately  preceding.  It  may  then  be  shown  as  an  ordinary 


H 


IT' 


FIG.  72. 


82 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


orthographic  projection,  just  as  though  the  original  horizontal 
plane  were  absent.  The  same  would  still  be  true  if  a  new  ver- 
tical plane  were  added  instead  of  the  horizontal  plane,  or,  if  an 
entirely  new  set  of  principal  planes  were  chosen  so  that  the 
new  principal  planes  would  be  parallel  to  the  old  ones. 

603.  Traces  of  planes  perpendicular  to  one  of  the  prin= 
cipal  planes.     Fig.  73  shows  a  plane  that  is  perpendicular  to 


the  horizontal  plane  but  inclined  to  the  vertical  plane.  It  may 
be  imagined  as  a  door  in  a  wall  of  a  room.  The  angle  with  the 
vertical  plane  (or  wall  in  this  case)  can  be  changed  at  will  by 
swinging  it  about  the  hinges,  yet  its  plane  always  remains  per- 


FIG.  74. 

pendicular  to  the  horizontal  plane  (floor  of  the  room).  The  inter- 
section with  the  vertical  plane  is  perpendicular  to  the  horizontal 
plane  because  it  is  the  intersection  of  two  planes  (the  given 
plane  and  the  vertical  plane),  each  of  which  is  perpendicular  to 
the  horizontal.  As  a  consequence  the  vertical  trace  is  perpen- 
dicular to  the  ground  line,  because,  when  a  line  is  perpendicular 


REPRESENTATION   OF  PLANES 


83 


to  a  plane  it  is  perpendicular  to  every  line  through  its  foot  (from 
geometry).  The  orthographic  representation  of  three  distinct 
planes  is  shown  on  the  right;  the  cases  selected  all  show  TV 
perpendicular  to  the  ground  line. 

What  is  true  regarding  planes  perpendicular  to  the  horizontal 
plane  is  equally  true  for  planes  similarly  related  with  respect  to  the 
vertical  plane.  Fig.  74  gives  an  illustration  of  such  a  case.  Here, 
the  horizontal  trace  is  perpendicular  to  the  ground  line  but  the 
vertical  trace  may  make  any  angle,  at  will,  with  the  ground  line. 

604.  Traces  of  planes  perpendicular  to  both  principal 
planes.  When  a  plane  is  perpendicular  to  both  principal  planes, 


-Y 


FIG.  75. 

its  two  traces  are  perpendicular  to  the  ground  line.     Such  a  plane 
is  a  profile  plane  and -is  shown  in  Fig.  75. 

605.  Traces  of  planes  inclined  to  both  principal  planes. 

It  may  be  inferred   from   the  preceding,  that,  if  the   plane   is 


FIG.  76. 

inclined  to  both  principal  planes,  neither  trace  can  be  perpen- 
dicular   to  the  ground  line.      Fig.  76   shows    such  a  case,  the 


84 


GEOMETRICAL  PROBLEMS  IN   PROJECTION 


orthographic  representation  of  which  in  projection  is  shown  on 
the  right. 

Fig.  77  shows  two  cards,  each  slit  half  way,  as  indicated. 
These  can  be  fitted  together  to  represent  the  principal  planes. 

If  at  any  point,  a  and  a',  on  XY 
slits  tt  and  ft'  be  made,  it  will 
be   found    on    assembling    the 
y          cards  that  a  third  card  can  be 

7^- Y    inserted. 

/(  In  the  upper  card,  another 

slit  sV  may  be  made,  through 
the  point  a',  with  its  direction 
parallel  to  tt.  As  in  the  previous 


FIG.  77. 


instance,  a  card  can  again  be 
inserted.  This  case  is  of  in- 
terest, however,  because  both  traces  become  coincident  on  the 
revolution  of  the  horizontal  plane,  and  fall  as  one  straight  line 
tTt',  as  shown  in  projection  in  Fig.  77. 

606.  Traces  of  planes  intersecting  the  ground  line.     It 

must  have  been  observed  in  the  cases  where  the  given  plane  is 
inclined  to  the  ground  line  that  both  traces  pass  through  the 
same  point  on  the  ground  line.  This  becomes  further  evident 
when  it  is  considered  that  the  ground  line  can  intersect  a  plane 
at  but  one  point,  if  at  all.  This  one  point  lies  in  the  ground  line, 
and,  hence,  it  has  coincident  projections  (509). 

607.  Plane  fixed  in  space  by  its  traces.     Two   intersect- 
ing lines  determine  a  plane  (from  solid  geometry).     Hence,  the 
two  traces  of  a  plane  fix  a  plane  with  reference  to  the  principal 
planes  because  the  traces  meet  at  the  same  point  on  the  ground 
line.     If  the  diagram  is  such  that  the  traces  do  not  intersect  in 
the  ground  line  within  the  limits  of  the  drawing,  it  is  assumed 
that  they  will  do  so  if  sufficiently  produced.     The  limiting  posi- 
tion of  a  plane  whose  traces  cannot  be  made  to  intersect  the 
ground  line  is  evidently  a  plane  parallel  to  it.     This  plane  is 
then  parallel  to  the  ground  line  and  its  traces  must  also  be  parallel 
to  it  (602). 

608.  Transfer    of     diagrams     from     orthographic     to 
oblique   projection.      Let    the    right-hand    diagram    of    Fig. 


REPRESENTATION  OF  PLANES 


85 


78  represent  a  plane  in  orthographic  projection.  Through 
any  point  S,  on  the  ground  line,  pass  a  profile  plane  sSs',  inter- 
secting the  two  traces  tT  and  Tt'  at  points  o  and  p.  To  transfer 
the  orthographic  to  the  oblique  projection,  lay  off  the  principal 
planes  HH  and  W  as  shown,  intersecting  in  the  ground  line 


FIG.  78. 

XY.  Lay  off  any  point  S  on  the  ground  line  in  the  oblique  pro- 
jection and  then  make  So  and  Sp  of  the  orthographic  equal  to 
the  similarly  lettered  lines  in  the  oblique  projection.  The  profile 
plane  sSs'  is  therefore  determined  in  the  oblique  projection. 
Make  TS  of  one  diagram  equal  to  the  TS  of  the  other,  and  com- 


FIG.  79. 

plete  by  drawing  the  traces  through  T,  o  and  T,  p.  To  increase 
the  clarity  of  the  diagram,  a  rectangular  plane  may  be  shown 
as  though  it  passes  through  the  principal  planes.  Fig.  78  has 
this  plane  added. 

A  case  where  the  two  traces  are  coincident  is  shown  in  Fig. 
79.     Two  profile  planes  are  required,  but  only  one  trace  of  each 


86  GEOMETRICAL  PROBLEMS   IN   PROJECTION 

is  needed.     All  the  necessary  construction  lines  are  shown  in  this 
diagram. 

609.  Traces  of  planes  in  all  angles.  As  planes  are  indef- 
inite in  extent,  so  are  their  traces;  and,  therefore,  the  traces  are 
.not  limited  to  any  one  angle.  In  the  discussion  of  most  problems, 
it  may  be  possible  to  choose  the  principal  planes  so  as  to  limit 
the  discussion  to  one  angle — usually  the  first.  The  advantage 
to  be  gained  thereby  is  the  greater  clarity  of  the  diagram,  as  then 
the  number  of  construction  lines  is  reduced  to  a  minimum.  Third 
angle  projection  may  also  be  used,  but  the  transfer  to  oblique 
projection  is  undesirable.  Second  and  fourth  angles  are  avoided 
because  the  projections  overlap  (317,  514,  516).  - 

Fig.  80  shows  the  complete  traces  of  a  given  plane  T.     The 


\ 


_,.  x-r/'  ~\7~  ^^  \.*  *  /  * 

A' 


/    X  T         V      XT/     V  T      1- 

X  \  ^       ~/i  2  8\        /4\  ^ 

\  A  t\'t      r* 

FIG.  80.  FIG.  81. 

full  lines  indicate  the  traces  in  the  first  angle;  the  dotted  lines 
show  the  continuation  in  the  remaining  three  angles.  Fig.  81 
represents  each  quadrant  separately  for  the  same  plane  that  is 
shown  in  Fig.  80.  The  appended  numbers  indicate  the  angle 
to  which  the  given  plane  is  limited. 

610.  Projecting  plane  of  lines.  It  is  now  evident  that 
the  finding  of  the  projection  of  a  line  is  nothing  more  nor 
less  than  the  finding  of  the  trace  of  its  projecting  piano.  The 
two  perpendiculars  from  a  line  to  the  plane  of  projection  are 
necessarily  parallel  and  therefore  determine  a  plane.  This  plane 
is  the  projecting  plane  of  the  line  and  its  intersection  (or 
trace)  with  the  plane  of  projection  contains  the  projection  of  the 
line. 

Manifestly,  any  number  of  lines  contained  in  this  projecting 
plane  would  have  the  same  projection  on  any  one  of  the  principal 
planes,  and  that  projection  therefore  does  not  fix  the  line  in  space. 


KEPRESENTATION  OF  PLANES 


87 


If  the  projection  on  the  corresponding  plane  of  projection  be  taken 
into  consideration,  the  projecting  planes  will  intersect,  and  this 


FIG.  82. 


intersection  will  be  the  given  line  in  space, 
the  point  in  question. 


Fig.  82  illustrates 


QUESTIONS  ON  CHAPTER  VI 

1.  What  is  the  trace  of  a  plane? 

2.  Draw  a  plane  parallel  to  the  vertical  plane  passing  through  the  first 

and  fourth  angles,  and  show  the  resulting  trace.     Make  diagram 
in  oblique  and  orthographic  projection. 

3.  Take  the  same  plane  of  Question  2  and  show  it  passing  through  the 

second  and  third  angles. 

4.  Show  how  a  plane  is  represented  when  it  is  parallel  to  the  horizontal 

plane  and  passes  through  the  first  and  second  angles.    Make 
diagram  in  oblique  and  orthographic  projection. 

5.  Take  the  same  plane  of  Question  4  and  show  the  trace  of  the  plane 

when  it  passes  through  the  third  and  fourth  angles. 

6.  Show  how  a  plane  is  represented  when  it  is  parallel  to  the  ground 

line  and  passes  through  the  first  angle. 

7.  How  is  a  plane  represented  when  it  is  parallel  to  the  ground  line  and 

passes  through  the  second  angle?     Third  angle?    Fourth  Angle? 

8.  Show  how  a  plane  passing  through  the  ground  line  is  indeterminate 

in  orthographic  projection. 

9.  When  a  plane  passes  through  the  ground  line  show  how  the  profile 

plane  might  be  used  to  advantage  in  representing  the  plane. 

10.  When  a  plane  passes  through  the  ground  line,  show  how  an  auxiliary 

principal  plane  may  be  used  to  obtain  determinate  traces. 

11.  When  a  plane  is  perpendicular  to  the  horizontal  plane  and  inclined  to 

the  vertical  plane,  show  how  this  is  represented  orthographically. 
Make,  also,  the  oblique  projection  of  it. 


88  GEOMETRICAL  PROBLEMS   IN   PROJECTION 

12.  When  a  plane  is  perpendicular  to  the  vertical  plane  and  inclined  to 

the  horizontal  plane  show  how  this  is  represented  orthographically. 
Make,  also,  the  oblique  projection  of  it. 

13.  When  a  plane  is  perpendicular  to  both  principal  planes  draw  the 

orthographic  traces  of  it. 

14.  Why  is  the  ground  line  perpendicular  to  both  traces  in  Question  13? 

Is  this  plane  a  profile  plane? 

15.  When  a  plane  is  inclined  to  the  ground  line  show  how  the  traces  are 

represented. 

16.  Why  do  both  traces  of  a  plane  intersect  the  ground  line  at  a  point 

when  the  plane  is  inclined  to  it? 

17.  Why  do  the  traces  fix  the  plane  with  reference  to  the  principal  planes? 


FIG.  6-A. 


18.  When  a  plane  is  parallel  to  the  ground  line,  why  are  the  traces  of  the 

plane  parallel  to  it? 

19.  Show  an  oblique  plane  in  all  four  angles  of  projection  (use  only  the 

limited  portion  in  one  angle  and  use  the  same  plane  as  in  the 
illustration). 

20.  In  what  angles  are  the  planes  whose  traces  are  shown  in  Fig.  6-A? 


t'.> 

.-.  / 

,x- 


t 
FIG.  6-B.  FIG.  6-C. 

21.  Show  how  a  line  in  space  and  its  projecting  perpendiculars  determine 

a  plane  which  is  the  projecting  plane  of  the  line.     Is  the  projection 
of  the  line  the  trace  of  the  projecting  plane  of  the  line? 

22.  Is  it  possible  to  have  two  separate  lines  whose  projections  are  coin- 

cident on  one  plane?     How  are  such  lines  determined? 

23.  Given  the  traces  of  a  plane  in  orthographic  projection  as  shown  in 

Fig.  6-B  construct  the  oblique  projection  of  it. 

24.  Construct  the  oblique  projection  of  a  plane  having  coincident  projec- 
tions (Fig.  6-C). 


CHAPTER  VII 
ELEMENTARY  CONSIDERATIONS  OF  LINES  AND  PLANES 

701.  Projection  of  lines  parallel  in  space.  When  two 
lines  in  space  are  parallel,  their  projecting  planes  are  parallel, 
and  their  intersection  with  any  third  plane  will  result  in  parallel 
lines.  If  this  third  plane  be  a  plane  of  projection,  then  the 
traces  of  the  two  projecting  planes  will  result  in  parallel  projections. 

Fig.  83  shows  two  lines,  AB  and  CD  in  space.     The  lines  are 


FIG.  83. 


FIG.  84. 


shown  by  their  horizontal  projections  ab  and  cd,  which  are  parallel 
to  each  other,  and  by  their  vertical  projections  a'b'  and  c'd', 
which  are  also  parallel  to  each  other.  A  perfectly  general  case 
is  represented  pictorially  on  a  single  plane  of  projection  in  Fig.  84. 


If  two 


702.  Projection  of  lines  intersecting  in  space. 

lines  in  space  intersect,  their  projections 
intersect,  because  the  two  lines  in  space  must 
meet  in  a  point.  Further,  the  projection  of 
this  point  must  be  common  to  the  projections 
of  the  lines. 

In  Fig.  85  two  such  lines,  AB  and  CB  are 
shown,  represented,  as  usual,  by  their  pro- 
jections.     O    is    the    intersection    in    space,  FIG.  85. 
indicated  by  its  horizontal  and  vertical  projections  o  and  o', 


90 


GEOMETRICAL  PROBLEMS  IN    PROJECTION 


respectively.  EF  and  GH  (Fig.  86)  are  two  other  lines,  chosen 
to  show  how  in  the  horizontal  plane  of  projection,  the  two  pro- 
jections may  coincide,  because  the  plane  of  the  two  lines  happens 
also  to  be  the  horizontal  projecting  plane.  The  case  is  not  inde- 
terminate, however,  as  the  vertical  projection  locates  the  point 
M  in  space.  The  reverse  of  this  is  also  true,  that  is,  the  vertical 


FIG.  86. 


FIG.  87. 


FIG.  88. 


instead  of  the  horizontal  projections  may  be  coincident.     General 
cases  of  the  above  are  represented  pictorially  in  Figs.  87  and  88. 
Should  the  horizontal  and  the  vertical  projections  be  coin- 
cident, the  lines  do  not  intersect  but  are  themselves  coincident 
in  space  and  thus  form  only  one  line. 

703.  Projection    of    lines    not  intersecting    in    space.* 

There  are  two  possible  cases  of  lines  that  do  not  intersect  in  space. 


FIG.  90. 


The  case  in  which  the  lines  are  parallel  to  each  other  has  previously 
been  discussed  (701).  If  the  two  lines  cannot  be  made  to  lie 
in  the  same  plane,  they  will  pass  each  other  without  intersecting. 
Hence,  if  in  one  plane  of  projection,  the  projections  intersect, 
they  cannot  do  so  in  the  corresponding  projection. 

This  fact  is  depicted  in  Fig.  89.    AB  and  CD  are  the  two 
*  Called  skew  lines. 


CONSIDERATIONS  OF  LINES  AND  PLANES 


91 


lines  in  space.  Two  distinct  points  E  and  F,  on  the  lines  are 
shown  in  the  horizontal  projection  as  e  and  f;  their  vertical 
projections  are,  however,  coincident.  Similarly,  G  and  H  are 
also  two  distinct  points  on  the  lines,  shown  as  g'  and  h'  in  the 
vertical  projection,  and  coincident,  as  g  and  h  in  the  horizontal 
projection.  The  pictorial  representation  is  given  in  Fig.  90. 

704.  Projection  of  lines  in  oblique  planes.  When  a 
third  plane  is  inclined  to  the  principal  planes,  it  cuts  them  in 
lines  of  intersection,  known  as  traces  (601).  Any  line,  when 
inclined  to  the  principal  planes  will  pierce  them  in  a  point.  Hence, 
if  a  plane  is  to  contain  a  given  line,  the  piercing  points  of  the 
line  must  lie  in  the  traces  of  the  plane.  Viewing  this  in  another 
way,  a  plane  may  be  passed  through  two  parallel  or  two  inter- 
secting lines.  On  the  resulting  plane,  any  number  of  lines  may 
be  drawn,  intersecting  the  given  pair.  Hence,  an  inclined  line 
must  pass  through  the  trace,  if  it  is  contained  in  the  plane.* 

Fig.  91  shows  a  plane  tit'  indicated  by  its  horizontal  trace 
tT  and  its  vertical  trace  TV. 
It  is  required  to  draw  a  line  AB 
in  this  plane.  Suppose  the  hori- 
zontal piercing  point  is  assumed 
at  b,  its  corresponding  vertical 
projection  will  lie  in  the  ground 
line  (509).  Also,  if  its  vertical 
piercing  point  is  assumed  at  a', 
its  corresponding  horizontal  pro- 
jection will  be  a.  With  two 
horizontal  projections  of  given 
points  on  a  line  and  two  vertical 

projections,  the  direction  of  the  line  is  determined,  for,  if  the 
horizontal  and  vertical  projecting  planes  be  erected,  their  inter- 
section determines  the  given  line  (610).f 

As  a  check  on  the  correctness  of  the  above,  another  line  CD 
may  be  assumed.  If  the  two  lines  lie  in  the  same  plane  and  are 
not  parallel,  they  must  intersect.  This  point  of  intersection 

*  When  the  plane  is  parallel  to  the  ground  line,  a  line  in  this  plane  parallel 
to  the  principal  planes  cannot  pierce  in  the  traces  of  the  plane.  See  Art.  602. 

f  It  must  be  remembered  that  the  principal  planes  must  be  at  right 
angles  to  each  other  to  determine  this  intersection.  In  the  revolved  position, 
the  planes  would  not  intersect  in  the  required  line. 


pIG 


92 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


is  M  shown  horizontally  projected  at  m  and  vertically,  at  m'. 
The  line  joining  the  two  is  perpendicular  to  the  ground  line  (508). 
This  is  illustrated  in  oblique  projection  in  Fig.  92. 

It  should  be  here  noted  that  if  the  line  is  to  be  contained  by 
the  plane,  only  the  direction  of  one  projection  can  be  assumed, 
and  that  the  corresponding  projection  must  be  found  by  the 
principles  so  far  developed. 

A  converse  of  this  problem  is  to  draw  a  plane  so  that  it  shall 
contain  a  given  line.  As  an  unlimited  number  of  planes  can  be 
passed  through  any  given  line,  the  direction  of  the  traces  is  not 
fixed.  Suppose  AB  (Fig.  91)  is  the  given  line,  then  through  the 
horizontal  piercing  point  b  draw  any  trace  tT,  and  from  T  where 
this  line  intersects  the  ground  line,  draw  Tt',  through  the  vertical 


FIG.  92. 

piercing  point  a'.  The  point  T  may  be  located  anywhere  along 
XY.  All  of  these  planes  will  contain  the  line  AB,  if  their  traces  pass 
through  the  piercing  points  of  the  line. 

Still  another  feature  of  Fig.  91  is  the  fact  that  from  it  can  be 
proved  that  two  intersecting  lines  determine  a  plane.  If  AB 
and  CD  be  the  two  lines  intersecting  at  M,  their  horizontal  piercing 
points  are  b  and  c  and  their  vertical  piercing  points  are  a!  and  d' 
respectively.  Two  points  fix  the  direction  of  a  line,  and,  hence, 
the  direction  of  the  traces  is  fixed;  tT  is  drawn  through  cb  and 
Tt'  is  drawn  through  a'd'.  The  check  lies  in  the  fact  that  both 
traces  intersect  at  one  point  T  on  the  ground  line  (606). 

705.  Projection  of  lines  parallel  to  the  principal  planes 
and  lying  in  an  oblique  plane.  If  a  line  is  horizontal,  it  is 
parallel  to  the  horizontal  plane,  and  its  vertical  projection  must  be 


CONSIDERATIONS  OF  LINES  AND  PLANES 


93 


parallel  to  the  ground  line  (511).  If  this  line  lies  in  an  oblique 
plane,  it  can  have  only  one  piercing  point  and  that  with  the  vertical 
plane,  since  it  is  parallel  to  the  horizontal.  Thus,  in  Fig.  93, 
let  tit'  be  the  given  oblique  plane,  represented,  of  course,  by  its 
traces.  The  given  line  is  AB  and  a'b'  is  its  vertical  projection, 
the  piercing  point  being  at  a'.  The  corresponding  horizontal 
projection  of  a'  is  a.  When  a  line  is  horizontal,  as  in  this  case, 
it  may  be  considered  as  being  cut  from  the  plane  fit  by  a  hori- 
zontal plane,  and,  as  such,  must  be  parallel  to  the  principal 
horizontal  plane.  These  two  parallel  horizontal  planes  are  cut 
by  the  given  oblique  plane  fit'  and,  from  geometry,  their  lines 
of  intersection  are  parallel.  Hence,  the  horizontal  projection 


.      FIG.  93. 

of  the  horizontal  line  must  be  parallel  to  the  horizontal  trace, 
of  the  given  plane  because  parallel  lines  in  space  have  parallel 
projections.  Accordingly,  from  a,  draw  ab,  parallel  to  Tt,  and 
the  horizontal  line  AB  is  thus  shown  by  its  projections. 

A  line,  parallel  to  the  vertical  plane,  drawn  in  an  oblique  plane 
follows  the  same  analysis,  and  differs  only  in  an  interchange  of 
the  operation.  That  is  to  say,  the  horizontal  projection  is  then 
parallel  to  the  ground  line  and  pierces  in  a  point  on  the  horizontal 
trace  of  the  given  plane;  its  vertical  projection  must  be  parallel 
to  the  vertical  trace  of  the  given  plane.  In  Fig.  93,  CD  is  a  line 
parallel  to  the  vertical  plane  and  cd  is  the  horizontal  projection, 
piercing  the  horizontal  plane  at  c  vertically  projected  at  c';  c'd' 
is,  therefore,  the  required  vertical  projection. 


94 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


A  check  on  the  problem  lies  in  the  fact  that  these  two  lines 
must  intersect  because  they  lie  in  the  same  plane  by  hypothesis. 
The  point  of  intersection  M  is  shown  as  m  in  the  horizontal 


FIG.  94., 

projection,  and  as  m'  in  the  vertical  projection.  The  line  joining 
these  points  is  perpendicular  to  the  ground  line.  The  oblique 
projection  of  this  is  shown  in  Fig.  94. 

Fig.  95  is  a  still  further  step  in  this  problem.     Three  lines 


FIG.  95. 


AB,  CD,  and  EF  are  shown,  all  being  in  the  plane  tTt'.  AB  is 
parallel  to  the  vertical  plane,  CD  is  parallel  to  the  horizontal 
plane  and  EF  is  any  other  line.  It  will  be  observed  that  the 
three  lines  intersect  so  as  to  form  a  triangle  MNO  shown  by 
having  the  area  shaded  in  both  its  projections. 


CONSIDERATIONS  OF  LINES  AND  PLANES 


95 


706.  Projections  of  lines  perpendicular  to  given  planes. 

If  a  line  is  perpendicular  to  a  given  plane,  the  projections  of  the 
line  are  perpendicular  to  the  corresponding  traces  of  the  plane. 

Let,  in  Fig.  96,  LL  be  any  plane  and  MM  any  other  plane 
intersecting  it  in  the  trace  tr.  Also,  let  AB  be  any  line,  perpen- 
dicular to  MM,  and  ab  be  the  projection  of  AB  on  LL.  It  is 
desired  to  show  that  the  projection  ab  is  perpendicular  to  the 
trace  tr.  Any  plane  through  AB  is  perpendicular  to  the  plane 
MM,  because  it  contains  a  line  perpendicular  to  the  plane  by 
hypothesis.  Also,  any  plane  through  Bb,  a  perpendicular  to 
the  plane  LL,  is  perpendicular  to  LL.  Hence,  any  plane  contain- 


FIG.  96. 

ing  both  lines  (it  can  do  so  because  they  intersect  at  B)  will  be 
perpendicular  both  to  LL  and  MM.  As  the  plane  through 
ABb  is  perpendicular  to  the  two  planes  LL  and  MM,  it  is  also 
perpendicular  to  a  line  common  to  the  two  planes,  such  as  tr. 
Thus,  ab  is  perpendicular  to  tr.  In  fact,  any  line  perpendicular 
to  a  plane  will  have  its  projection  on  any  other  plane  perpendicular 
to  the  trace  of  the  plane,  because,  instead  of  LL  being  assumed 
as  the  plane  of  projection  and  BA  a  perpendicular  to  another  plane 
MM,  the  conditions  may  be  reversed  and  MM  be  assumed  as 
the  plane  of  projection  and  Bb  the  perpendicular. 

The  converse  of  this  is  also  true.  If  Ao  be  assumed  the  pro- 
jection of  Bb  on  MM,  a  plane  passed  through  the  line  Ao  per- 
pendicular to  the  trace  tr  will  contain  the  lines  AB  and  Bb,  where 


96 


GEOMETEICAL  PROBLEMS   IN   PROJECTION 


Bb  is  the  given  line  and  AB  then  the  projecting  perpendicular 
to  the  plane  MM. 

707.  Revolution  of  a  point  about  a  line.  Frequently  it 
is  desirable  to  know  the  relation  of  a  point  with  respect  to  a  line, 
for,  if  the  line  and  point  are  given  by  their  projections,  the  true 
relation  may  not  be  apparent.  To  do  this,  revolve  the  point 
about  the  line  so  that  a  plane  through  the  point  and  the  line 
will  either  coincide  or  be  parallel  to  the  plane  of  projection,  then 
they  will  be  projected  in  their  true  relation  to  each  other.  The 
actual  distance  between  the  point  and  the  line  is  then  shown 
as  the  perpendicular  distance  from  the  point  to  the  line. 

Consider  the  diagram  in  Fig.  97*  and  assume  that  the  point 
A  is  to  be  revolved  about  the  point  B.  The  projection  of  A  on 


B         «         a" 


FIG.  97/ 


the  line  Oa"  is  a  and,  to  an  observer  looking  down  from  above 
the  line  Oa",  the  apparent  distance  between  A  and  B  is  aB.  Oa' 
is  the  apparent  distance  of  A  from  B  to  an  observer,  looking 
orthographically,  from  the  right  at  a  plane  perpendicular  to 
the  plane  of  the  paper  through  Oa'.  Neither  projection  gives 
the  true  relation  between  A  and  B  from  a  single  projection.  If 
the  point  A  is  revolved  about  B  as  an  axis,  with  BA  as  a  radius, 
until  it  coincides  with  the  line  Oa",  A  will  either  be  found  at  a" 
or  at  a'",  depending  upon  the  direction  of  rotation.  During  the 
revolution,  A  always  remains  in  the  plane  of  the  paper  and  de- 
scribes a  circle,  the  plane  of  which  is  perpendicular  to  the  axis, 
through  B,  the  centre  of  the  circle. 

*  This  diagram  is  a  profile  plane  of  the  given  point  and  of  the  principal 


CONSIDERATIONS  OF  LINES  AND  PLANES 


97 


A  more  general  case  is  shown  in  Fig.  98  where  BB  is  an  axis 
lying  in  a  plane,  and  A  is  any  point  in  space  not  in  the  plane 
containing  BB.  If  A  be  revolved  about  BB  as  an  axis,  it  will 
describe  a  circle,  the  plane  of  which  will  be  perpendicular  to 
the  axis.  In  other  words,  A  will  fall  somewhere  on  a  line  Ba" 
perpendicular  to  BB.  The  line  Ba"  must  be  perpendicular  to 
BB  because  it  is  the  trace  of  a  perpendicular  plane  (706).  This 
point  is  a"  and  Ba"  is  equal  to  the  radius  BA.  Contrast  this 


FIG.  98. 

with  a,  the  orthographic  projection  of  A  on  the  plane  containing 
BB.  Evidently,  then,  a  is  at  a  lesser  distance  from  BB  than  a". 
Indeed,  BA  equals  Ba"  and  is  equal  to  the  hypothenuse  of  a 
right  triangle,  whose  base  is  the  perpendicular  distance  of  the 
projection  of  the  point  from  the  axis,  and  whose  altitude  is  the 
distance  of  the  point  above  the  plane  containing  the  line.  The 
angle  AaB  is  a  right  angle,  because  a  is  the  orthographic  projection 
of  A. 

QUESTIONS  ON  CHAPTER  VII 

1.  If  two  lines  in  space  are  parallel,  prove  that  their  projections  on  any 

plane  are  parallel. 

2.  When  two  lines  in  space  intersect,  prove  that  their  projections,  on 

any  plane,  intersect. 

3.  Draw  two  lines  in  space  that  are  not  parallel  and  still  do  not  intersect. 

4.  Show  a  case  of  two  non-intersecting  lines  whose  horizontal  projections 

are  parallel  to  each  other  and  whose  vertical  projections  intersect. 
Show  also,  that  the  horizontal  projecting  planes  of  these  lines  are 
parallel. 


98  'GEOMETRICAL  PROBLEMS   IN   PROJECTION 

5.  Make  an  oblique  projection  of  the  lines  considered  in  Question  4. 

6.  Prove  that  when  a  line  lies  in  a  plane  it  must  pass  through  the  traces 

of  the  plane. 

7.  Given  one  projection  of  a  line  in  a  plane,  find  the  corresponding  pro- 

•  jection. 

8.  Given  a  plane,  draw  intersecting  lines  in  the  plane  and  show  by  the 

construction  that  the  point  of  intersection  satisfies  the  ortho- 
graphic representation  of  a  point. 

9.  Show  how  two  intersecting  lines  determine  a  plane  by  aid  of  an 

oblique  projection. 

10.  In  a  given  oblique  plane,  draw  a  line  parallel  to  the  horizontal  plane 

and  show  by  the  construction  that  this  line  pierces  the  vertical 
plane  only.  Give  reasons  for  the  construction. 

11.  In  Question  9,  draw  another  line  parallel  to  the  vertical  plane  and 

show  that  this  second  line  intersects  the  first  in  a  point. 

12.  Given  an  oblique  plane,  draw  three  lines;  one  parallel  to  the  hori- 

zontal plane,  one  parallel  to  the  vertical  plane  and  the  last  inclined 
to  both  planes.  Show,  by  the  construction,  that  the  three  lines 
form  a  triangle  (or  meet  in  a  point  in  an  exceptional  case). 

13.  Prove  that  when  a  line  is  perpendicular  to  a  plane  the  projection  of 

this  line  on  any  other  plane  is  perpendicular  to  the  trace  of  the 
plane.  Show  the  general  case  and  also  an  example  in  orthographic 
projection. 

14.  A  line  lies  in  a  given  plane  and  a  point  is  situated  outside  of  the 

plane.  Show  how  the  point  is  revolved  about  the  line  until  it  is 
contained  in  the  plane. 

15.  Prove  that  the  point,  while  revolving  about  the  line,  describes  a 

circle  the  plane  of  which  is  perpendicular  to  the  axis  (the  line  about 
which  it  revolves). 


CHAPTER  VIII 

V 

PROBLEMS  INVOLVING  THE  POINT,  THE  LINE,  AND  THE  PLANE 

801.  Introductory.     A  thorough  knowledge  of  the  preceding 
three  chapters  is  necessary  in  order  to  apply  the  principles,  there 
developed,  in  the  solution  of  certain  problems.     The  commercial 
application  of  these  problems  frequently  calls  for  extended  knowl- 
edge in  special   fields  of   engineering,  and  for  this  reason,  the 
application,  in  general,  has  been  avoided. 

Countless  problems  of  a  commercial  nature  may  be  used  as 
illustrations.  All  of  these  indicate,  in  various  ways,  the  im- 
portance of  the  subject.  In  general,  the  commercial  problem 
may  always  be  reduced*  to  one  containing  the  mathematical 
essentials  (reduced  to  points,  lines  and  planes).  The  solution, 
then,  may  be  accomplished  by  the  methods  to  be  shown  sub- 
sequently. 

802.  Solution  of  problems.     In  the  solution  of  the  following 
problems,  three  distinct  steps  may  be  noted:  the  statement,  the 
analysis,  and  the  construction. 

The  statement  of  the  problem  gives  a  clear  account  of  what 
is  to  be  done  and  includes  the  necessary  data. 

The  analysis  entails  a  review  of  the  principles  involved,  and 
proceeds,  logically,  from  the  given  data  to  the  required  conclusion. 
On  completion  of  the  analysis*  the  problem  is  solved  to  all  intents 
and  purposes. 

The  construction  is  the  graphical  presentation  of  the  analysis. 
It  is  by  means  of  a  drawing  and  its  description  that  the  giver 
data  is  associated  with  its  solution.  It  may  be  emphasized 
again  that  the  drawings  are  made  orthographically,  and  that  the 
actual  points,  lines  and  planes  are  to  be  imagined. 

By  a  slight  change  in  the  assumed  data,  the  resultant  con- 
struction may  appear  widely  different  from  the.  diagrams  in  the 
book.  Here,  then,  is  an  opportunity  to  make  several  con- 
structions for  the  several  assumptions,  and  to  prove  that  all 

99 


100  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

follow  the  general  analysis.  The  simpler  constructions  might 
be  taken  and  transformed  from  orthographic  to  oblique  projec- 
tion; this  will  show  the  projections  as  well  as  the  actual  points, 
lines  and  planes  in  space.  By  performing  this  transformation 
(from  orthographic  to  oblique  projection),  the  student  will  soon 
be  able  to  picture  the  entire  problem  in  space,  without  recourse 
to  any  diagrams. 

To  bring  forcibly  to  the  student's  attention  the  difference 
between  the  analysis  and  the  construction,  it  may  be  well  to 
note  that  the  analysis  gives  the  reasoning  in  its  most  general 
terms,  while  the  construction  is  specific,  in  so  far  as  it  takes  the 
assumed  data  and  gives  the  solution  for  that  particular  case  only. 

803.  Problem  1.  To  draw  a  line  through  a  given  point,  parallel 
to  a  given  line. 

Analysis.     If  two  lines  in  space  are  parallel,  their  projecting 
planes  are  parallel  and  their  intersection  with  the  principal  planes 
are  parallel  (701).     Hence,  through  the  projections  of  the  given 
point,  draw  lines  parallel  to  the  projections  of  the  given  lines. 
Construction.     Let  AB,  Fig.  99,  be  the  given  line  in  space, 
d,      represented   by  its  horizontal  pro- 
jection ab  and  its  vertical  projection 
a'b'.     Further,   let  G  be  the  given 
point,   similarly  represented   by  its 
horizontal  projection  g  and  its  ver- 
tical   projection   g'.     Through  the 
horizontal    projection    g,  draw    cd 
parallel  to  ab  and  through  g',  draw 
FlG  99>  c'd'  parallel  to  a'b'. 

As  the  length  of  the  line  is  not 

specified,  any  line  that  satisfies  the  condition  of  parallelism  is 
permissible.  Therefore,  CD  is  the  line  in  space  that  is  parallel 
to  AB  through  the  point  G.  A  pictorial  representation  of  this 
is  shown  in  Fig.  84. 

804.  Problem  2.    To  draw  a  line  intersecting  a  given  line 
at  a  given  point. 

Analysis.  If  two  lines  in  space  intersect,  they  intersect  in 
a  point  that  is  common  to  the  two  lines.  Therefore,  their  pro- 
jecting planes  will  intersect  in  a  line  which  is  the  projecting  line 
of  the  given  point  (702).  Hence,  through  the  projections  of  the 


THE  POINT,   THE  LINE,  AND  THE  PLANE 


101 


FIG.  100. 


given  point,  draw  any  lines,  intersecting  the  projections  of  the 
given  lines. 

Construction.  Let  AB,  Fig.  100,  be  the  given  line,  shown 
horizontally  projected  as  ab  and  ver- 
tically projected  as  a'b'.  Let,  also, 
G  be  the  given  point,  situated  on  the 
line  AB.  As  no  direction  is  specified 
for  the  intersecting  line,  draw  any 
line  cd  through  the  horizontal  projec- 
tion g  and  this  will  be  the  horizontal 
projection  of  the  required  line. 
Similarly,  any  other  line  c'd'  through 
g'  will  be  the  vertical  projection  of 
the  required  line.  Hence,  CD  and 

AB  are  two  lines  in  space,  intersecting  at  the  point  G.      The 
pictorial  representation  of  this  case  is  depicted  in  Figs.  87  and  88. 

805.  Problem  3.    To  find  where  a  given  line  pierces  the 
principal  planes. 

Analysis.  If  a  line  is  oblique  to  the  principal  planes,  it  will 
pierce  each  of  these  in  a  point,  the  corresponding  projection  of 
which  will  be  in  the  ground  line.  Hence,  a  piercing  point  in  any 
principal  plane  must  be  on  the  projection  of  the  line  in  that 
plane.  It  must  also  be  on  a  perpendicular  erected  at  the  point 
where  the  corresponding  projection  crosses  the  ground  line. 
Therefore,  the  required  piercing  point  is  at  their  intersection. 
Construction.  Let  AB,  Fig.  101,  be  a  limited  portion  of  an 
indefinite  line,  shown  by  its  horizontal 
projection  ab  and  its  vertical  projection 
a'b'.  The  portion  chosen  AB  will  not 
pierce  the  principal  planes,  but  its  continua- 
tion, in  both  directions,  will.  Prolong  the 
vertical  projection  a'b'  to  c'  and  at  c',  erect 
a  perpendicular  to  XY,  the  ground  line,  and 
continue  it,  until  it  intersects  the  prolonga- 
tion of  ab  at  c.  This  will  be  the  horizontal 
piercing  point.  In  the  same  way,  prolong 
ab  to  d,  at  d,  erect  a  perpendicular  to  the 
ground  line  as  dd',  the  intersection  of  which  with  the  prolonga- 
tion of  a'b',  at  d'  will  give  the  vertical  piercing  point.  Hence, 


FIG.  101. 


102 


GEOMETRICAL  PROBLEMS  IN   PROJECTION 


if  CD  be  considered  as  the  line,  it  will  pierce  the  horizontal  plane 
of  projection  at  c  and  the  vertical  plane  of  projection  at  d'.  A  pic- 
torial representation  in  oblique  projection  is  shown  in  Fig.  52. 

806.  Problem  4.  To  pass  an  oblique  plane,  through  a  given 
oblique  line. 

Analysis.  If  a  plane  is  oblique  to  the  principal  planes,  it 
must  intersect  the  ground  line  at  a  point  (606);  and  if  it  is  to 
contain  a  line,  the  piercing  points  of  the  line  must  lie  in  the  traces 
of  the  plane  (704).  Therefore,  to  draw  an  oblique  plane  containing 
a  given  oblique  line,  join  the  piercing  points  of  the  line  with  any 
point  of  the  ground  line  and  the  resulting  lines  will  be  the  traces 
of  the  required  plane. 

Construction.     Let  AB,  Fig.  102,  be  the  given  line.     This 


FIG.  102. 


FIG.  103. 


line  pierces  the  horizontal  plane  at  a  and  the  vertical  plane  at  b'. 
Assume  any  point  T  on  the  ground  line  XY,  and  join  T  with  a 
and  also  with  b'.  Ta  and  Tb'  are  the  required  traces,  indicated, 
as  usual,  by  tTt'.  Thus,  the  plane  T  contains  the  line  AB.  The 
construction  in  oblique  projection  is  given  in  Fig.  103. 

807.  Special  cases  of  the  preceding  problem.     If    the 

line  is  parallel  to  both  planes  of  projection,  the  traces  of  the  plane 
will  be  parallel  to  the  ground  line  (602),  and  the  profile  plane 
may  be  advantageously  used  in  the  drawing. 

If  the  line  is  parallel  to  only  one  of  the  principal  planes,  join 
the  one  piercing  point  with  any  point  on  the  ground  line,  which 
results  in  one  trace  of  the  required  plane.  Through  the  point 
on  the  ground  line,  draw  the  corresponding  trace,  parallel  to  the 


THE  POINT,   THE  LINE,  AND  THE  PLANE 


103 


projection  of  the  line  in  the  plane  containing  this  trace.     The 
case  is  evidently  that  considered  in  Art.  705. 

808.  Problem  5.  To  pass  an  oblique  plane,  through  a  given 
point. 

Analysis.  If  the  oblique  plane  is  to  contain  a  given  point,  it 
will  also  contain  a  line  through  the  point.  Hence,  through  the 
given  point,  draw  an  oblique  line  and  find  the  piercing  points 
of  this  line  on  the  principal  planes.  Join  these  piercing 
points  with  the  ground  line  and  the  result  will  be  the  required 
traces  of  the  plane. 

Construction.  Let  C,  Fig.  104,  be  the  required  point,  and 
let  AB  be  a  line  drawn  through  this  point.  AB  pierces  the  hori- 


FIG.  104. 


FIG.  105. 


b    H 


zontal  plane  at  a  and  the  vertical  plane  at  b'.  Join  a  and  b' 
with  any  assumed  point  T  and  tTt'  will  be  the  required  traces. 
The  plane  T,  contains  the  point  C,  because  it  contains  a  line  AB 
through  the  point  C.  Fig.  105  shows  this  pictorially. 

NOTE.  An  infinite  number  of  planes  may  be  passed  througri 
the  point,  hence  the  point  T  was  assumed.  It  might  have  been 
assumed  on  the  opposite  side  of  the  point  and  would  still  have 
contained  the  given  point,  or  the  auxiliary  line  through  the 
point. 

Also,  it  is  possible  to  draw  a  line  through  the  given  point 
parallel  to  the  ground  line.  A  profile  construction  will  then  be 
of  service  (602). 

Again,  a  perpendicular  line  may  be  drawn  through  the  given 
point  and  a  plane  be  passed  so  that  it  is  perpendicular  to  one  or 
both  planes  of  projection  (603,  604). 


104 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


809.  Problem  6.  To  find  the  intersection  of  two  planes, 
oblique  to  each  other  and  to  the  principal  planes. 

Analysis.  If  two  planes  are  oblique  to  each  other,  they 
intersect  in  a  line.  Any  line  in  a  plane  must  pass  through  the 
traces  of  the  plane.  As  the  line  of  intersection  is  common  to  the 
two  planes,  it  must  pass  through  the  traces  of  both  planes  and 
hence  it  passes  through  the  intersection  of  these  traces. 

Construction.  Let  T  and  S,  Fig.  106,  be  the  given  planes. 
The  horizontal  piercing  point  of  their  line  of  intersection  is  at  b, 


FIG.  106. 


FIG.  107. 


vertically  projected  at  b';  the  vertical  piercing  point  is  at  a', 
horizontally  projected  at  a.  Join  ab  and  aV,  as  they  are  the 
projections  of  the  line  AB,  which  is  the  intersection  of  the  planes 
T  and  S.  In  oblique  projection,  this  appears  as  shown  in  Fig.  107. 

810.  Special  case  of  the  preced= 
ing  problem.  If  the  two  planes  are 
chosen  so  that  the  traces  in  one  plane 
do  not  intersect  within  the  limits  of 
the  drawing,  then  draw  an  auxiliary 
plane  R  (Fig.  108)  and  find  the  inter- 
section, AB  as  shown.  From  c'  draw 
c'd',  parallel  to  a'b'  and  from  c,  draw 
cd,  parallel  to  ab.  The  line  of  inter- 
section of  the  given  planes  is  thus  deter- 
mined. 


FIG.  108. 


811.  Problem  7.  To  find  the  corresponding  projection  of  a 
given  point  lying  in  a  given  oblique  plane,  when  one  of  its  pro- 
jections is  given. 

Analysis.      If  a  line  lies  in  a  given  plane  and  also  contains 


THE  POINT,   THE  LINE,  AND  THE  PLANE 


105 


a  given  point,  the  projections  of  this  line  will  also  contain  the 
projections  of  the  given  point.  Hence,  through  the  projection 
of  the  given  point,  draw  the  projection  of  a  line  lying  in  the  given 
plane.  Then  find  the  corresponding  projection  of  the  line. 
The  required  projection  of  the  given  point  will  lie  on  the  inter- 
section of  a  perpendicular  to  the  ground  line,  through  the  given 
projection  of  the  point  with  the  corresponding  projection  of  the, 
line. 

Construction.  Let  tTt',  Fig.  109,  be  the  traces  of  the  given 
plane  and  c  the  horizontal  projection  of  the  given  point.  Draw 
ab,  the  horizontal  projection  of  a  line  in  the  plane,  through  c 
the  horizontal  projection  of  the  given  point.  The  horizontal 


FIG.  109. 


FIG.  110. 


piercing  point  of  the  line  AB  is  in  the  trace  Tt  at  a,  and  its  cor- 
responding projection  lies  in  the  ground  line  at  a'.  Further, 
the  vertical  piercing  point  lies  on  a  perpendicular  to  the  ground 
line  from  the  point  b  and  also  in  the  trace  Tt',  hence,  it  is  at  b' 
and  a'b'  is  thus  the  corresponding  projection  of  the  line  ab.  The 
required  projection  of  the  given  point  lies  on  a  line  through 
c,  perpendicular  to  the  ground  line,  and  also  on  a'b';  hence, 
it  is  at  their  intersection  c'.  The  point  C,  in  space,  is  con- 
tained in  the  plane  T,  and  c  and  c',  are  corresponding  pro- 
jections. The  oblique  projection  of  this  problem  is  given  in 
Fig.  110. 

812.  Special  case  of  the  preceding  problem.  The  point 
in  the  above  problem  was  purposely  chosen  in  the  first  angle,  in 
order  to  obtain  a  simple  case.  It  may  be  located  any  where ,. 


106  GEOMETRICAL  PROBLEMS  IX  PROJECTION 

however,  because  the  planes  are  indefinite  in  extent.  For  instance, 
in  Fig.  Ill  the  vertical  projection  is  selected  below  the  ground 
line.  However,  a  single  projection  does  not  locate  a  point  in 
space.  It  may  be  assumed  as  lying  either  in  the  third  or  fourth 
angles  (515).  Subsequent  operations  are  dependent  upon  the 

angle  in  which  the  point  is  chosen. 
Assume,  for  instance,  that  the  point 
is  in  the  fourth  angle;  the  traces  of 
the  given  plane  must  then  also  be 
assumed  as  being  in  the  fourth  angle. 
Thus,  with  the  given  plane  T,  Tt  is 
the  horizontal  trace,  and  Tt'"  is  the 
vertical  trace  (609).  In  completing 
pIG  in  the  construction  by  the  usual  method, 

let  c'  be  the  assumed  vertical  pro- 
jection, and  through  it,  draw  a'b'  as  the  vertical  projection  of 
the  assumed  line  through  the  given  point  and  lying  in  the  given 
plane.  This  line  pierces  the  horizontal  plane  at  b  and  the  vertical 
plane  at  a'.  Hence,  ab  and  a'b'  are  the  corresponding  projections 
of  the  line  AB  in  space  which  is  situated  in  the  fourth  angle. 
The  corresponding  horizontal  projection  of  c'  is  c,  and  thus  the 
point  C  in  space  is  determined. 

Had  the  point  been  assumed  in  the  third  angle,  then  the 
traces  Tt"  and  Tt'"  would  have  been  the  ones  to  use,  Tt"  being 
the  horizontal,  and  Tt'"  the  vertical  trace.  The  construction 
would  then,  in  general,  be  the  same  as  the  previous. 

813.  Problem  8.  To  draw  a  plane  which  contains  a  given 
point  and  is  parallel  to  a  given  plane. 

Analysis.  The  traces  of  the  required  plane  must  be  parallel 
to  the  traces  of  the  given  plane.  A  line  may  be  drawn  through 
the  given  point  parallel  to  an  assumed  line  in  the  given  plane. 
This  line  will  then  pierce  the  principal  planes  in  the  traces  of  the 
required  plane.  Hence,  in  the  given  plane,  draw  any  line. 
Through  the  given  point,  draw  a  line  parallel  to  it,  and  find  the 
piercing  points  of  this  line  on  the  principal  planes.  Through 
these  piercing  points  draw  the  traces  of  the  required  plane  parallel 
to  the  corresponding  traces  of  the  given  plane.  The  plane  so 
drawn  is  parallel  to  the  given  plane. 

Construction.     Let  T,  Fig.  112  be  the  given  plane,  and  G 


THE   POINT,   THE  LINE,  AND  THE  PLANE 


107 


the  given  point.  In  the  plane  T,  draw  any  line  CD  as  shown  by 
cd  and  c'd'  its  horizontal  and  vertical  projections  respectively. 
Through  G,  draw  AB,  parallel  to  CD  and  ab  and  a'b'  will  be 
the  projections  of  this  line.  The  piercing  points  are  a  and  b' 
on  the  horizontal  and  vertical  planes  respectively.  Draw  b'S 
parallel  to  t'T  and  aS  parallel  to  tT,  then  sSs'  will  be  the  traces 


FIG.  113. 


of  the  required  plane,  parallel  to  the  plane  T  and  containing  a 
given  point  G.  A  check  on  the  accuracy  of  the  construction  is 
furnished  by  having  the  two  traces  meet  at  S.  Also,  only  one 
plane  will  satisfy  these  conditions  because  the  point  S  cannot 
be  selected  at  random.  This  construction  is  represented  picto- 
rially  in  Fig.  113. 

814.  Problem  9.    To  draw  a  line  perpendicular  to  a  given 
plane  through  a  given  point. 

Analysis.  If  a  line  is  perpendicular 
to  a  given  plane  the  projections  of  the 
line  are  perpendicular  to  the  corresponding 
traces  of  the  plane.  Hence,  draw  through 
each  projection  of  the  given  point,  a 
line  perpendicular  to  the  corresponding  trace 
(706). 

Construction.  Let  T,  Fig.  114,  be 
the  given  plane,  and  C,  the  given  point. 
Through  c,  the  horizontal  projection  of  the  given  point, 
draw  ab,  perpendicular  to  Tt;  and  through  c',  the  vertical  pro- 
jection of  the  given  point,  draw  a'b',  perpendicular  to  Tt'. 


FIG.  114. 


108 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


Thus,  AB  is  perpendicular  to  the  plane  T. 
of  this  problem  is  given  in  Fig.  115. 


An  oblique  projection 


815.  Special    case   of   the   preceding   problem.      If   the 

point  is  chosen  in  the  third  angle,  then  it  must  be  observed  that 
Tt"  is  the  horizontal  trace  (Fig.  116)  and  Tt'"  is  the  vertical 
trace.  Again,  the  line  AB  is  drawn  perpendicular  to  T,  by 
making  a'b',  through  c',  perpendicular  to  TV",  and  ab,  through  c, 
perpendicular  to  Tt". 

816.  Problem  10.     To  draw  a  plane  through  a  given  point 
perpendicular  to  a  given  line. 

Analysis.     The  traces  of  the  required  plane  must  be  perpen- 


a'  >* 


FIG.  115. 


FIG.  116. 


dicular  to  the  corresponding  projections  of  the  given  line  (706), 
Through  one  projection  of  the  given  point,  draw  an  auxiliary  line 
parallel  to  the  trace;  the  corresponding  projection  of  this  line 
will  be  parallel  to  the  ground  line,  because  it  is  a  line  parallel  to 
that  plane  in  which  its  projection  is  parallel  to  the  trace 
(705).  Find  the  piercing  point  of  this  auxiliary  line,  and 
through  this  point,  draw  a  line  perpendicular  to  the  corre- 
sponding projection  of  the  given  line.  Where  this  intersects 
the  ground  line,  draw  another  line,  perpendicular  to  the 
corresponding  projection  of  the  given  line.  The  traces  are  thus 
determined. 

Construction.  Let,  in  Fig.  117,  AB  be  the  given  line,  and 
C  the  given  point.  For  convenience,  assume  a  horizontal  line 
in  the  plane  as  the  auxiliary  line.  Then,  through  c,  draw  cd, 


THE  POINT,  THE  LINE,   AND  THE  PLANE 


109 


perpendicular  to  ab,  and  through  c',  draw  c'd',  parallel  to  the 
ground  line.  The  piercing  point  of  this  auxiliary  line  is  d',  and 
only  one  point  on  either  trace  is  required.  Hence,  through  d', 
draw  TV,  perpendicular  to  a'b',  and  from  T,  draw  Tt,  per- 
pendicular to  ab.  T  is,  therefore,  the  required  plane,  and  Tt 
must  be  parallel  to  cd  because  both  must  be  perpendicular  to 
ab.  Fig.  118  shows  a  pictorial  representation  of  the  same 
problem. 

NOTE.  Instead  of  having  assumed  a  line  parallel  to  the 
horizontal  plane,  a  line  parallel  to  the  vertical  plane  might  have 
been  assumed.  In  the  latter  case,  the  vertical  projection  would 
have  been  perpendicular  to  the  vertical  projection  of  the  line, 


FIG.  118. 


and  the  horizontal  projection  would  therefore  have  been  parallel 
to  the  ground  line.  Also,  a  point  in  the  horizontal  plane  would 
have  fixed  the  traces,  instead  of  a  point  in  the  vertical  plane  as 
shown  in  the  problem. 

817.  Problem  11.  To  pass  a  plane  through  three  given 
points  not  in  the  same  straight  line. 

Analysis.  If  two  of  the  points  be  joined  by  a  line,  a  plane 
may  be  passed  through  this  line  and  revolved  so  that  it  contains 
the  third  point.  In  this  position,  the  plane  will  contain  a  line 
joining  the  third  point  with  any  point  on  the  first  line.  Hence, 
join  two  points  by  a  line,  and  from  any  point  on  this  line,  draw 
another  line,  through  the  remaining  point.  Find  the  piercing 
points  of  these  two  lines,  and  thus  establish  the  traces  of  the 
required  plane. 


110 


GEOMETRICAL  PROBLEMS   IN  PROJECTION 


Construction.  Let  AB  and  C,  in  Fig.  119,  be  the  three  given 
points.  Join  A  and  B  and  find  where  this  line  pierces  the  prin- 
cipal planes  at  d  and  e'.  Assume  any  point  H,  on  the  line  AB, 
and  join  H  and  C;  this  line  pierces  the  principal  planes  at  g  and 
f.  Join  dg  and  f e'  and  the  traces  obtained  are  those  of  the 
required  plane.  A  check  on  the  accuracy  of  the  work  is  furnished 
by  the  fact  that  both  traces  must  meet  at  one  point  on  the  ground 
line,  as  shown  at  T.  Hence,  the  plane  T  contains  the  points 
A,  B  and  C.  Fig.  120  is  an  oblique  projection  of  this  problem. 

NOTE.  In  the  construction  of  this  and  other  problems,  it 
may  be  desirable .  to  work  the  problem  backwards  in  order  to 


FIG.  119. 


FIG.  120. 


obtain  a  simpler  drawing.  It  is  quite  difficult  to  select  three 
points  of  a  plane,  at  random,  so  that  the  traces  of  the  plane 
shall  meet  the  ground  line  within  the  limits  of  the  drawing.  In 
working  the  problem  backwards,  the  traces  are  first  assumed, 
then,  any  two  distinct  lines  are  drawn  in  the  plane,  and,  finally, 
three  points  are  selected  on  the  two  assumed  lines  of  the  plane. 
It  is  good  practice,  however,  to  assume  three  random  points 
and  proceed  with  the  problem  in  the  regular  way.  Under  these 
conditions,  the  piercing  points  are  liable  to  be  in  any  angle, 
and,  as  such,  furnish  practice  in  angles  other  than  the  first. 

818.  Problem  12.  To  revolve  a  given  point,  not  in  the 
principal  planes,  about  a  line  lying  in  one  of  the  principal  planes. 

Analysis.  If  a  point  revolves  about  a  line,  it  describes  a 
circle,  the  plane  of  which  is  perpendicular  to  the  axis  of  revolu- 


THE  POINT,  THE  LINE,   AND  THE  PLANE  111 

tion.  As  the  given  line  is  the  axis,  the  point  will  fall  somewhere 
in  the  trace  of  a  plane,  through  the  point,  perpendicular  to  the 
axis.  The  radius  of  the  circle  is  the  perpendicular  distance  from 
the  point  to  the  line;  and  is  equal  to  the  hypothenuse  of  a  tri- 
angle, whose  base  is  the  distance  from  the  projection  of  the  point 
to  line  in  the  plane,  and  whose  altitude  is  the  distance  of  the 
corresponding  projection  of  the  point  from  the  plane  containing 
the  line  (707). 

Construction.  Assume  that  the  given  line  AB,  Fig.  121, 
lies  in  the  horizontal  plane  and  therefore  is  its  own  projection, 
ab,  in  that  plane;  its  corresponding  projection  is  a'b'  and  lies 
in  the  ground  line.  Also,  let  C  be  the  , 

given  point,   shown  by  its   projections  c  CK 

and  c'.     Through  c,  draw   cp,  perpendic- 
ular to  the   line  ab;  cp  is  then  the  trace     v  «_'          &' 
of  the   plane  of   the  revolving  point  and 


the  revolved  position  of  the  point  will  fall 


somewhere  along  this  line.     The  radius  of 


p\ 


\\ 


the  circle  is  found  by  making  an  auxiliary  \c" 

view,  in  which  c'o  is  the  distance  of  the  pIG  12i. 

point    above    the    horizontal   plane,   and 

oq  =  cp  is  the  distance  of  the  horizontal  projection  of  the  given 
point  from  the  axis.  Hence,  c'q  is  the  radius  of  the  circle,  and, 
therefore,  lay  off  pc"  =  c'q.  The  revolved  position  of  the  point 
C  in  space  is,  therefore,  c".  The  distance  pc"  might  have  been 
revolved  in  the  opposite  direction  and  thus  would  have  fallen 
on  the  opposite  side  of  the  axis.  This  is  immaterial,  however, 
as  the  point  is  always  revolved  so  as  to  make  a  clear  diagram. 
NOTE.  In  the  construction  of  this  problem,  the  line  was 
assumed  as  lying  in  the  horizontal  plane.  It  might  have  been 
assumed  as  lying  in  the  vertical  plane,  and,  in  this  case,  .the 
operations  would  have  been  identical,  the  only  difference  being 
that  the  trace  of  the  plane  containing  the  path  of  the  point  would 
then  lie  in  the  vertical  plane.  The  auxiliary  diagram  for  deter- 
mining the  radius  of  the  circle  may  be  constructed  below  the 
ground  line,  or,  if  desirable,  in  an  entirely  separate  diagram. 

819.  Problem  13.    To  find  the  true  distance  between  two 

points  in  space  as  given  by  their  projections.    First  method. 

Analysis.     The  true  distance  is  equal  to  the  length  of  the 


112 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


line  joining  the  two  points.  If,  then,  one  projecting  plane  of 
the  line  be  revolved  until  it  is  parallel  to  the  corresponding  plane 
of  projection,  the  line  will  be  shown  in  its  true  length  on  the 
plane  to  which  it  is  parallel. 

Construction.  Case  1.  When  both  points  are  above  the 
plane  of  projection. — Let  AB,  in  Fig.  122,  be  the  given  line. 
For  convenience,  revolve  the  horizontal  projecting  plane  about 
the  projecting  perpendicular  from  the  point  A  on  the  line.  The 
point  B  will  describe  a  circle,  the  plane  of  which  is  perpendicular 
to  the  axis  about  which  it  revolves.  As  the  plane  of  the  circle 
(or  arc)  is  parallel  to  the  horizontal  plane,  it  is  projected  as 
the  arc  be.  The  corresponding  projection  is  bV,  because  its 


&'       c 


FIG.  122. 


FIG.  123. 


plane  is  perpendicular  to  the  vertical  plane.  In  the  position 
ac,  the  projecting  plane  of  AB  is  parallel  to  the  vertical  plane, 
and  c'  is  the  vertical  projection  of  the  point  b,  when  so 
revolved.  Hence,  a'c'  is  the  true  length  of  the  line  AB  in 
space. 

820.  Case  2.  When  the  points  are  situated  on  opposite  sides 
of  the  principal  plane. — Let  AB,  in  Fig.  123,  be  the  given  line. 
Revolve  the  horizontal  projecting  plane  of  the  line  about  the 
horizontal  projecting  perpendicular  from  the  point  A  on  the 
line.  The  point  B  will  describe  an  arc  which  is  horizontally  pro- 
jected as  be  and  vertically  projected  as  b'c'.  The  ultimate 
position  of  b'  after  revolution  is  at  c',  whereas  a'  remains  fixed. 
Hence,  a'c'  is  the  true  length  of  the  line  AB. 


THE  POINT,  THE   LINE,  AND  THE  PLANE 


113 


821.  Problem  13.    To  find  the  true  distance  between  two 
points  in  space  as  given  by  their  projections.     Second  method. 

Analysis.  The  true  distance  is  equal  to  the  length  of  a 
line  joining  the  two  points.  If  a  plane  be  passed  through  the 
line  and  revolved  about  the  trace  into  one  of  the  principal  planes, 
the  distance  between  the  points  will  remain  unchanged,  and  in 
its  revolved  position,  the  line  will  be  shown  in  its  true  length. 

Construction.  Case  1.  When  both  points  are  above  the 
plane  of  projection. — Let  A  and  B  Fig.  124  be  the  two  points  in 
question.  For  convenience,  use  the  hori- 
zontal projecting  plane  of  the  line  as  the 
revolving  plane;  its  trace  on  the  horizontal 
plane  is  ab,  which  is  also  the  projection 
of  the  line.  The  points  A  and  B,  while 
revolving  about  the  line  ab,  will  describe 
circles,  the  planes  of  which  are  perpendic- 
ular to  the  axis,  and,  hence,  in  their  re- 
volved  position,  will  lie  along  lines  aa" 
and  bb".  In  this  case,  the  distance  of  the 
projections  of  the  points  from  the  axis  is 
zero,  because  the  projecting  plane  of  the 
line  joining  the  points  was  used.  The 

altitudes  of  the  triangles  are  the  distances  of  the  points  above 
the  horizontal  plane,  and,  hence,  they  are  also  the  hypothenuses 
of  the  triangles  which  are  the  radii  of  the  circles.*  Therefore, 
lay  off  aa"  =  a'o  and  bb//  =  b/p  along  lines  aa"  and  bb",  which 
are  perpendicular  to  ab.  Hence,  a"b"  is  the  true  distance 
between  the  points  A  and  B  in  space. 

822.  Case  2.    When   the   points    are    situated    on   opposite 

sides  of  the  principal  plane. — Let  AB, 
Fig.  125,  be  the  two  points.     From  the 
-y    projections  it  will  be  seen  that  A  is  in 
the  first   angle  and  B  is  in  the  fourth 
angle.     If,  as  in  the  previous  case,  the 
horizontal    projecting  plane  is  used  as 
the  revolving  plane  then  the  horizontal 
FlG  125  projection  ab  is  the  trace  of  the  revolv- 

ing plane  as  before.     The  point  A  falls 

to  a"  where  aa"  =  a'o.     Similarly,  B  falls  to  b"  where  bb"  =  b'p, 
*  See  the  somewhat  similar  case  in  Problem  12. 


b" 

A 


114 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


but,  as  must  be  noticed,  this  point  falls  on  that  side  of  ab,  oppo- 
site to  the  point  a".  A  little  reflection  will  show  that  such  must 
be  the  case,  but  it  may  be  brought  out  by  what  follows :  The 
line  AB  pierces  the  horizontal  plane  at  m  and  this  point  must 
remain  fixed  during  the  revolution.  That  it  does,  is  shown  by 
the  fact  that  the  revolved  position  of  the  line  a"b"  passes 
through  this  point. 

NOTE.  In  both  cases,  the  vertical  projecting  plane  might 
have  been  used  as  the  revolving  plane  and  after  the  operation 
is  performed,  the  true  length  of  the  line  is  again  obtained.  It 
must  of  necessity  be  equal  to  that  given  by  the  method  here 
indicated. 

823.  Problem  14.  To  find  where  a  given  line  pierces  a 
given  plane. 

Analysis.     If    an    auxiliary    plane    be    passed   through    the 


FIG.  126. 


FIG.  127. 


given  line,  so  that  it  intersects  the  given  plane,  it  will  cut  from 
it  a  line  that  contains  the  given  point.  The  given  point  must 
also  lie  on  the  given  line,  hence  it  lies  on  their  intersection. 

Construction.  Let  T,  Fig.  126,  be  the  given  plane  and  AB 
the  given  line.  For  convenience,  use  the  horizontal  projecting 
plane  of  the  line  as  the  auxiliary  plane;  the  horizontal  trace  is 
cb  and  the  vertical  trace  is  cc'.  The  auxiliary  plane  cuts  from 
the  plane  T  the  line  CD.  The  vertical  projection  is  shown  as 
c'd'  and  the  horizontal  projection  cd  is  contained  in  the  trace 
of  the  horizontal  projecting  plane,  because  that  plane  was  pur- 


THE  POINT,  THE  LINE,  AND  THE  PLANE 


115 


posely  taken  as  the  cutting  plane  through  the  line.  It  is  only  in 
the  vertical  projection  that  the  intersection  m'  is  determined; 
its  horizontal  projection  m  is  indeterminate  in  that  plane  (except 
from  the  fact  that  it  is  a  corresponding  projection)  since  the 
given  line  and  the  line  of  intersection  have  the  same  horizontal 
projecting  plane.  An  oblique  projection  of  this  problem  is  given 
in  Fig.  127. 

824.  Problem  15.  To  find  the  distance  of  a  given  point 
from  a  given  plane. 

Analysis.  The  perpendicular  distance  from  the  given  point 
to  the  given  plane  is  the  required  distance.  Hence,  draw  a  per- 
pendicular from  the  given  point  to  the  given  plane,  and  find 
where  this  perpendicular  pierces  the  .given  plane.  If  the  line 
joining  the  given  point  and  the  piercing  point  be  revolved  into 
one  of  the  planes  of  projection,  the  line  will  be  shown  in  its  true 
length. 

Construction.  Let  T,  Fig.  128,  be  the  given  plane,  and  A, 
the  given  point.  From  A,  draw  AB, 
perpendicular  to  the  plane  T;  the  pro- 
jections of  AB  are  therefore  perpendicu- 
lar to  the  traces  of  the  plane.  If  the 
horizontal  projecting  plane  of  the  line 
AB  be  used  as  the  auxiliary  plane,  it 
cuts  the  given  plane  in  the  line  CD, 
and  pierces  it  at  the  point  B.  Revolve 
the  projecting  plane  of  AB  about  its 
horizontal  trace  ab  into  the  horizontal 
plane  of  projection.  A  will  fall  to  a", 
where  aa"  =  a'o,  and  B  will  fall  to  b", 
where  bb"  =  b'p.  Therefore,  a"b"  is  the 
distance  from  the  point  A  to  the  plane  T. 


FIG.  128. 


825.  Problem  16.  To  find  the  distance  from  a  given  point 
to  a  given  line. 

Analysis.  Through  the  given  point  pass  a  plane  perpen- 
dicular to  the  given  line.  The  distance  between  the  piercing 
point  of  the  given  line  on  this  plane  and  the  given  point  is  the 
required  distance.  Join  these  two  points  by  a  line  and  revolve 
this  line  into  one  of  the  planes  of  projection;  the  line  will  then 
be  seen  in  its  true  length. 


116 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


FIG.  129. 


Construction.  Let  AB,  Fig.  129,  be  the  given  line,  and  G, 
the  given  point.  Through  G,  draw  a  plane  perpendicular  to  AB 
(816),  by  drawing  gc  perpendicular  to  ab,  and  g'c',  parallel  to 

the  ground  line;  the  piercing  point 
of  this  line  on  the  vertical  plane 
is  c'.  Hence,  the  traces  Tt'  and 
Tt,  perpendicular  to  a'b'  and  ab, 
respectively,  through  the  point  c', 
will  be  the  traces  of  the  required 
plane.  AB  pierces  this  plane  at 
B,  found  by  using  the  horizontal 
projecting  plane  of  AB  as  a  cutting 
plane,  this  cutting  plane  inter- 
secting in  a  line  DE.  The  point 
B  must  be  on  both  AB  and  DE. 
The  projected  distance,  then,  is 
the  distance  between  the  points  B 
and  G ;  the  true  distance  is  found 

by  revolving  BG  into  the  horizontal  plane.  The  latter  operation 
is  accomplished  by  using  the  horizontal  projecting  plane  of  BG 
and  revolving -it  about  its  trace  bg;  G  falls  to  g",  where  gg"  = 
g'p,  and  B  falls  to  b",  where  bb"  =  b'o.  Thus,  g"b"  is  the  true 
distance  between  the  point  G  and  the  line  AB. 

826.  Problem  17.  To  find  the  angle  between  two  given 
intersecting  lines. 

Analysis.  If  a  plane  be  passed  through  these  lines  and 
revolved  into  one  of  the  planes  of  projection,  the  angle  will  be 
shown  in  its  true  size.  Hence,  find  the  piercing  points  of  the 
given  lines  on  one  of  the  planes  of  projection;  the  line  joining 
these  piercing  points  will  be  the  trace  of  the  plane  containing 
the  lines.  Revolve  into  that  plane  and  the  revolved  position 
of  the  two  lines  shows  the  true  angle. 

Construction.  Let  AB  and  AC,  Fig.  130,  be  the  two  given 
lines  intersecting  at  A.  These  lines  pierce  the  horizontal  plane 
of  projection  at  b  and  c  and  be  is  the  trace  of  a  plane  containing 
the  two  lines.  If  A  be  considered  as  a  point  revolving  about 
the.  line  be,  it  then  describes  a  circle,  the  plane  of  which  is  per- 
pendicular to  be  and  the  point  A  will  coincide  with  the  horizontal 
plane  somewhere  along  the  line  oa".  The  radius  of  the  circle 


THE  POINT,   THE   LINE,  AND  THE  PLANE  117 

described  by  A  is  equal  to  the  hypothenuse  of  a  right  triangle, 
where  the  distance  ao,  the  projec- 
tion of  a  from  the  axis,  is  the  base, 
and  a'p,  the  distance  of  the  point 
above  the  plane,  is  the  altitude. 
This  is  shown  in  the  triangle  a'pq, 
where  pq  is  equal  to  ao  and  therefore 
a'q  is  the  required  radius.  Hence, 
make  oa"  =  a'q,  and  a"  is  the  re- 
volved position  of  the  point  A  in 
space.  The  piercing  points  b  and  c 
of  the  given  lines  do  not  change 

their    relative    positions.     Thus  a"b 

...  ,      ,          ...  FIG.  130. 

and  a  c  are   the   revolved    position 

of  the  given  lines,  and  the  angle  ba"c  is  the  true  angle. 

827.  Problem  18.  To  find  the  angle  between  two  given 
planes. 

Analysis.  If  a  plane  be  passed  perpendicular  to  the  line  of 
intersection  of  the  two  given  planes,  it  will  cut  a  line  from  each 
plane,  the  included  angle  of  which  will  be  the  true  angle.  Re- 
volve this  plane,  containing  the  lines,  about  its  trace  on  the 
principal  plane,  until  it  coincides  with  that  plane,  and  the  angle 
will  be  shown  in  its  true  size. 

Construction.  Let  T  and  S,  Fig.  131,  be  the  two  given 
planes,  intersecting,  as  shown,  in  the  line  AB.  Construct  a  sup- 
plementary plane  to  the  right  of  the  main  diagram.  H'H'  is 
the  new  horizontal  plane,  shown  as  a  line  parallel  to  ab.  The 
line  AB  pierces  the  vertical  plane  at  a  distance  a'a  above  the 
horizontal  plane.  Accordingly,  a'a  is  laid  off  on  V'V,  perpendicu- 
lar to  H'H' ;  it  also  pierces  the  horizontal  plane  at  b  shown  in  both 
views.  The  supplementary  view  shows  ba'  in  its  true  relation 
to  the  horizontal  plane,  and  is  nothing  more  or  less  than  a  side 
view  of  the  horizontal  projecting  plane  of  the  line  AB.  If,  in 
this  supplementary  view,  a  perpendicular  plane  cd  be  drawn, 
it  will  intersect  the  line  AB  in  c,  and  the  horizontal  plane  in  the 
trace  dfe  shown  on  end.  The  lettering  in  both  views  is  such 
that  similar  letters  indicate  similar  points.  Hence,  dfe  is  the 
trace  of  the  plane  as  shown  in  the  main  diagram,  and  ec  and 
dc  are  the  two  lines  cut  from  the  planes  T  and  S  by  the  plane 


118 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


cde.  When  the  plane  cde  is  revolved  into  coincidence  with 
the  horizontal  plane,  c  falls  to  c"  in  the  supplementary  view 
and  is  projected  back  to  the  main  diagram  as  c".  Therefore, 
ec"d  is  the  true  angle  between  the  planes,  because  e  and  d  remain 
fixed  in  the  revolution. 

828.  Problem  19.  To  find  the  angle  between  a  given  plane 
and  one  of  the  principal  planes. 

Analysis.  If  an  auxiliary  plane  be  passed  through  the  given 
plane  and  the  principal  plane  so  that  the  auxiliary  plane  is  per- 


FIG.  131. 

pendicular  to  the  intersection  of  the  given  plane  and  the  prin- 
cipal plane,  it  will  cut  from  each  a  line,  the  included  angle  of 
which  will  be  the  true  angle.  If,  then,  this  auxiliary  plane  be 
revolved  into  the  principal  plane,  the  angle  will  be  shown  in  its 
true  size. 

Construction.  Let  T,  Fig.  132,  be  the  given  plane.  The 
angle  that  this  plane  makes  with  the  horizontal  plane  is  to  be 
determined.  Draw  the  auxiliary  plane  R,  so  that  its  horizontal 
trace  is  perpendicular  to  the  horizontal  trace  of  the  given  plane; 
the  vertical  trace  of  the  auxiliary  plane  must  as  a  consequence 
be  perpendicular  to  the  ground  line  as  Rr'.  A  triangle  rRr'  is 


THE  POINT,  THE  LINE,  AND  THE  PLANE 


119 


cut  by  the  auxiliary  plane  from  the  given  plane  and  the  two 
principal  planes.  If  this  triangle  be  revolved  into  the  horizontal 
plane,  about  rR  as  an  axis,  the  point  r'  will  fall  to  r"  with  Rr' 
as  a  radius.  Also,  the  angle  rRr"  must  be  a  right  angle,  because 
it  is  cut  from  the  principal  planes,  which  are  at  right  angles  to 
each  other.  Hence,  Rrr"  is  the  angle  which  the  plane  T  makes 
with  the  horizontal  plane  of  projection. 

The  construction  for  obtaining  the  angle  with  the  vertical 
plane  is  identical,  and  is  shown  on  the  right-hand  side,  with 
plane  S  as  the  given  plane.  All  construction  lines  are  added 
and  no  comment  should  be  necessary. 

NOTE.  The  similarity  of  Probs.  18  and  19  should  be  noted. 
In  Prob.  19  the  horizontal  trace  is  the  intersection  of  the  given 


' 


s/« 


FIG.  132. 


plane  and  the  horizontal  plane;  hence,  the  auxiliary  plane  is 
passed  perpendicular  to  the  trace.  A  similar  reasoning  applies  to 
the  vertical  trace. 

829.  Problem  20.  To  draw  a  plane  parallel  to  a  given  plane, 
at  a  given  distance  from  it. 

Analysis.  The  required  distance  between  the  two  planes 
is  the  perpendicular  distance,  and  the  resulting  traces  must  be 
parallel  to  the  traces  of  the  given  plane.  If  a  plane  be  passed 
perpendicular  to  either  trace,  it  will  cut  from  the  principal  planes 
and  the  given  plane  a  right  angled  triangle,  the  hypothenuse 
of  which  will  be  the  line  cut  from  the  given  plane.  If,  further, 
the  triangle  be  revolved  into  the  plane  containing  the  trace  and 
the  required  distance  between  the  planes  be  laid  off  perpendicular 
to  the  hypothenuse  cut  from  the  given  plane,  it  will  establish 
a  point  on  the  hypothenuse  of  the  required  plane.  A  line  parallel 


120  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

to  the  hypothenuse  through  the  established  point  will  give  the 
revolved  position  of  a  triangle  cut  from  the  required  plane.  On 
the  counter  revolution,  this  triangle  will  determine  a  point  in 
each  plane,  through  which  the  required  traces  must  pass.  Hence, 
if  lines  be  drawn  through  the  points  so  found,  parallel  to  the 
traces  of  the  given  plane,  the  traces  of  the  required  plane  are 
established. 

Construction.  Let  T,  Fig.  133,  be  the  given  plane,  and  r"g 
the  required  distance  between  the  parallel  planes.  Pass  a  plane 

rOr',  perpendicular  to  the  hori- 
zontal trace  Tt;  its  vertical  trace 
Or  is,  therefore,  perpendicular 
to  the  ground  line.  The  re- 
volved position  of  the  triangle 
cut  from  the  two  principal  planes 
and  the  given  plane  is  rOr". 
Lay  off  r"g,  perpendicular  to 
rr",  and  equal  to  the  required 
distance  between  the  planes. 
Draw  uu"  parallel  to  rr"  and 

FlG  j33  the  triangle  cut  from  the  prin- 

cipal planes  and  the  required 
plane  is  obtained.  On  counter  revolution,  u"  becomes  u'  and  u 
remains  fixed;  Su'  and  Su,  parallel  respectively  to  Tr'  and  1*r, 
are  the  required  traces.  Therefore,  the  distance  between  the 
planes  T  and  S  is  equal  to  r"g. 

830.  Problem  21.    To  project  a  given  line  on  a  given  plane. 

Analysis.  If  perpendiculars  be  dropped  from  the  given 
line  upon  the  given  plane,  the  points,  so  found,  are  the  projec- 
tions of  the  corresponding  points  on  the  line.  Hence,  a  line 
joining  the  projections  on  the  given  plane  is  the  required  pro- 
jection of  the  line  on  that  plane. 

Construction.  Let  T,  Fig.  134,  be  the  given  plane,  and 
AB  the  given  line.  From  A,  draw  a  perpendicular  to  the  plane 
T;  its  horizontal  projection  is  ac  and  its  vertical  projection  is 
a'c'.  To  find  where  AC  pierces  the  given  plane,  use  the  hori- 
zontal projecting  plane  of  AC  as  the  cutting  plane;  FE  is  the 
line  so  cut,  and  C  is  the  resultant  piercing  point.  Thus,  C  is 
the  projection  of  A  on  the  plane  T.  A  construction,  similar  in 


THE  POINT,  THE  LINE,  AND  THE  PLANE 


121 


detail,  will  show  that  D  is  the  projection  of  B  on  the  plane  T. 
Hence,  CD  is  the  projection  of  AB  on  the  plane  T. 

831.  Problem  22.  To  find  the  angle  between  a  given  line 
and  a  given  plane. 

Analysis.  The  angle  made  by  a  given  line  and  a  given  plane 
is  the  same  as  the  angle  made  by  the  given  line  and  its  projection 


FIG.  134. 

on  that  plane.  If  from  any  point  on  the  given  line,  another 
line  be  drawn  parallel  to  the  projection  on  the  given  plane,  it 
will  also  be  the  required  angle.  The  projection  of  the  given  line 
on  the  given  plane  is  perpendicular  to  a  projecting  perpendicular 
from  the  given  line  to  the  given  plane.  Hence,  any  line,  parallel 
to  the  projection  and  lying  in  the  projecting  plane  of  the  given 
line  to  the  given  plane  is  also  perpendicular  to  this  projecting 
perpendicular.  Therefore,  pass  a  plane  through  the  given  line 
and  the  projecting  perpendicular  from  the  given  line  to  the 
given  plane.  Revolve  this  plane  into  one  of  the  planes  of  pro- 


122 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


jection,  and,  from  any  point  on  the  line,  draw  a  perpendicular 
to  the  projecting  perpendicular.  The  angle  between  this  line 
and  the  given  line  is  the  required  angle. 

Construction.  Let  T,  Fig.  135,  be  the  given  plane,  and 
AB  the  given  line.  From  B,  draw  a  perpendicular  to  the  plane 
T,  by  making  the  projections  respectively  perpendicular  to  the 
traces  of  the  given  plane.  Find  the  piercing  points  e  and  f, 
on  the  horizontal  plane,  of  the  given  line  and  this  perpendicular. 
Revolve  the  plane  containing  the  lines  BE  and  BF;  B  falls  to  b", 
on  a  line  b"p,  perpendicular  to  ef.  The  distance  b"p  is  equal 
to  the  hypothenuse  of  a  right  triangle,  where  bp  is  the  base  and 


FIG.  135. 


b'o  is  the  altitude;  b'oq  is  such  a  triangle,  where  oq  =  bp.  Hence, 
b"p  is  laid  off  equal  to  b'q.  If  from  any  point  d,  a  line  dc  be 
drawn,  perpendicualr  to  b"f,  then  b"dc  is  the  required  angle, 
as  dc  is  parallel  to  the  projection  of  AB  on  the  plane  T  in  its 
revolved  position. 

832.  Problem  23.  To  find  the  shortest  distance  between 
a  pair  of  skew  *  lines. 

Analysis.  The  required  line  is  the  perpendicular  distance 
between  the  two  lines.  If  through  one  of  the  given  lines,  another 
line  be  drawn  parallel  to  the  other  given  line,  the  intersecting  lines 
will  establish  a  plane  which  is  parallel  to  one  of  the  given  lines. 

*  Skew  lines  are  lines  which  are  not  parallel  and  which  do  not  intersect. 


THE   POINT,  THE  LINE,  AND  THE  PLANE 


123 


The  length  of  a  perpendicular  from  any  point  on  the  one  given 
line  to  the  plane  containing  the  other  given  line,  is  the  required 
distance. 

Construction.    Let  AB  and  CD,  Fig.  136,  be  the  two  given 


FIG.  136. 

lines.  Through  any  point  O,  on  AB,  draw  FE,  parallel  to  CD, 
and  determine  the  piercing  points  of  the  lines  AB  and  FE;  a, 
e  and  f,  b'  are  these  piercing  points,  and,  as  such,  determine 
the  plane  T.  CD  is  then  parallel  to  the  plane  T.  From  any 
point  G,  on  CD,  draw  GH,  perpendicular  to  the  plane  T,  and 


124 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


find  its  piercing  point  on  that  plane.  This  point  is  H,  found 
by  drawing  gh  perpendicular  to  Tt  and  g'h'  perpendicular  to 
Tt';  the  horizontal  projecting  plane  cuts  from  the  plane  T,  a 
line  MN,  on  which  is  found  H,  the  piercing  point.  GH  is  there- 
fore the  required  distance,  but  to  find  its  true  length,  revolve 


FIG.  .136. 

the  horizontal  projecting  plane  of  GH  into  the  horizontal  plane. 
On  revolution,  H  falls  to  h",  where  hh"  is  equal  to  the  distance 
h'  above  the  ground  line;  and,  similarly,  G  falls  to  g".  There- 
fore, h"g"  is  the  true  distance  between  the  lines  AB  and  CD. 


THE  POINT,  THE  LINE,  AND  THE  PLANE 


125 


NOTE.  In  order  to  find  the  point  on.  each  of  the  lines  at 
which  this  perpendicular  may  be  drawn,  project  the  one  given 
line  on  the  plane  containing  the  other.  Where  the  projections 
cross,  the  point  will  be  found. 

ADDITIONAL  CONSTRUCTIONS 

833.  Application  to  other  problems.  The  foregoing  prob- 
lems may  be  combined  so  as  to  form  additional  ones.  In  such 
cases,  the  analysis  is  apt  to  be  rather  long,  and  in  the  remaining 
few  problems  it  has  been  omitted.  The  construction  of  the 


FIG.  137. 

problem  might  be  followed  by  the  student  and  then  an  analysis 
worked  up  for  the  particular  problem  afterward. 

834.  Problem  24.  Through  a  given  point,  draw  a  line  of  a 
given  length,  making  given  angles  with  the  planes  of  projection. 

Construction.  Consider  the  problem  solved,  and  let  AB, 
Fig.  137,  be  the  required  line,  through  the  point  A.  The  con- 
struction will  first  be  shown  and  then  the  final  position  of  the 
line  AB  will  be  analytically  considered.  From  a',  draw  a'c' 
of  a  length  1,  making  the  angle  a  with  the  ground  line,  and  draw 
ac,  parallel  to  the  ground  line.  The  horizontal  projecting  plane 
of  AB  has  been  revolved  about  the  horizontal  projecting  perpen- 
dicular, until  it  is  parallel  to  the  vertical  plane,  and,  therefore 


126 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


a'c'  is  shown  in  its  true  length  and  inclination  to  the  horizontal 
plane.  On  the  counter-revolution  of  the  projecting  plane  of  AB, 
it  will  be  observed  that  the  point  A  remains  fixed,  because  it 
lies  in  the  axis;  B  describes  a  circle,  however,  whose  plane  is 
parallel  to  the  horizontal  plane,  and  is,  therefore,  projected  as 
the  arc  cb,  while  its  vertical  projection  (or  trace,  if  it  be  consid- 
ered as  a  plane  instead  of  the  moving  point)  is  b'c',  a  line  parallel 
to  the  ground  line.  The  angle  that  the  line  AB  makes  with  the 
horizontal  plane  is  now  fixed,  but  the  point  B  is  not  finally  located 
as  the  remaining  condition  of  making  the  angle  ji  with  the  hori- 
zontal plane  is  yet  conditional. 

In  order  to  lay  off  the  angle  that  the  line  makes  with  the 


FIG.  137. 


vertical  plane,  draw  a  line  ad,  through  a,  making  the  angle  (} 
with  the  ground  line,  and  of  a  length  1.  From  a',  draw  a'd', 
so  that  d'  is  located  from  its  corresponding  projection  d.  Draw 
bd,  through  d,  parallel  to  the  ground  line,  and  where  its  inter- 
section with  the  arc  cb,  locates  b,  the  final  position  of  the  hori- 
zontal projection  of  the  actual  line.  The  corresponding  pro- 
jection b',  may  be  located  by  drawing  the  arc  dV,  and  finding 
where  it  intersects  the  line  b'c',  through  c',  parallel  to  the  ground 
line.  The  points  b  and  b'  should  be  corresponding  projections, 
if  the  construction  has  been  carried  out  accurately.  In  reviewing 
the  latter  process,  it  is  found  that  the  vertical  projecting  plane 


THE  POINT,  THE  LINE,  AND  THE  PLANE 


127 


of  the  line  has  been  revolved  about  the  vertical  projecting  per- 
pendicular through  A,  until  it  was  parallel  to  the  horizontal 
plane.  The  horizontal  projection  is  then  ad,  and  this  is  shown 
in  its  true  length  and  inclination  to  the  vertical  plane. 

It  may  also  be  noted  that  the  process  of  finding  the  ultimate 
position  of  the  line  is  simply  to  note  the  projections  of  the  path 
of  the  moving  point,  when  the  projecting  planes  of  the  line  are 
revolved.  That  is,  when  the  horizontal  projecting  plane  of  the 
line  is  revolved,  the  line  makes  a  constant  angle  with  the  hori- 
zontal plane;  and  the  path  of  the  moving  point  is  indicated  by 
its  projections.  Similarly,  when  the  vertical  projecting  plane 
of  the  line  is  revolved,  the  line  makes  a  constant  angle  with  the 
vertical  plane;  and  the  path  of  the  moving  point  is  again  given 


b 

1  <4  a', 

6' 


FIG.  138. 

by  its  projections.  Where  the  paths  intersect,  on  the  proper 
planes,  it  is  evidently  the  condition  that  satisfies  the  problem. 
There  are  four  possible  solutions  for  any  single  point  in  space, 
and  they  are  shown  in  Fig.  138.  Each  is  of  the  required  length, 
and  makes  the  required  angles  with  the  planes  of  projection. 
The  student  may  try  to  work  out  the  construction  in  each  case 
and  show  that  it  is  true. 

835.  Problem  25.  Through  a  given  point,  draw  a  plane, 
making  given  angles  with  the  principal  planes. 

Construction.  Prior  to  the  solution  of  this  problem,  it  is 
desirable  to  investigate  the  property  of  a  line  from  any  point 
on  the  ground  line,  perpendicular  to  the  plane.  If  the  angles 
that  this  line  makes  with  the  principal  planes  can  be  determined, 
the  actual  construction  of  it,  in  projection,  is  then  similar  to  the 
preceding  problem.  To  draw  the  required  plane,  hence,  resolves 
itself  simply  into  passing  a  plane  through  a  given  point,  per- 
pendicular to  a  given  line  (816). 


128 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


Let,  in  Fig.  139,  T  be  the  required  plane,  making  the  required 
angles  a  with  the  horizontal  plane,  and  g  with  the  vertical  plane. 
To  find  the  angle  made*  with  the  horizontal  plane,  pass  a  plane 
perpendicular  to  the  horizontal  trace  as  AOB,  through  any  as- 
sumed point  O  on  the  ground  line;  OA  is4  thus  perpendicular 
to  Tt  and  OB  is  perpendicular  to  the  ground  line.  This  plane 
cuts  from  the  plane  T,  a  line  AB,  and  the  angle  BAO  is  the  required 
angle  a.  Similarly,  pass  a  plane  COD,  through  O  perpen- 
dicular to  the  vertical  trace,  then  CO  is  perpendicular  to  Tt 
and  OD  is  perpendicular  to  the  ground  line;  DCO  is  the  required 
angle  g,  which  the  plane  T  makes  with  the  vertical  plane.  The 
planes  AOB  and  COD  intersect  in  a  line  OP.  OP  is  perpendicular 


FIG.  139. 

to  the  plane  T  because  each  plane  AOB  and  COD  is  perpendicular 
to  the  plane  T  (since  they  are  each  perpendicular  to  a  trace, 
which  is  a  line  in  the  plane),  and,  hence,  the  line  common  to  the 
two  planes  (OP)  must  be  perpendicular  to  the  plane.  The 
angles  OPC  and  OPA  are,  therefore,  right  angles,  and,  as  a  result, 
angle  POA  =  90°-«,  and  angle  POC  =  90°-g.  Hence,  to 
draw  a  perpendicular  to  the  required  plane,  draw  a  line  making 
angles  witH  the  principal  planes  equal  to  the  complements  (90° 
vininus  the  angle)  of  the  corresponding  angles.  It  is  evident 
that  any  line  will  do,  as  all  •  such  lines,  when  measured  in  the 
same  way,  will  be  parallel,  hence,  it  is  not  necessary,  although 
convenient,  that  this  line  should  pass  through  the  ground  line. 
To  complete  the  problem,  le't  AB,  Fig.  140,  be  a  line  making 


THE  POINT,  THE  LINE,  AND  THE  PLANE 


129 


angles  90° -a  with  the  horizontal  plane,  and  90° -&  with  the 
vertical  plane.  Through  G,  the  given  point,  draw  a  perpendicular 
plane  to  AB,  and  the  resultant  plane  T  is.  the  required  plane 


FIG.  140. 

making  an  angle  a,  with  the  horizontal  plane,  and  an  angle  (i, 
with  the  vertical  plane. 

NOTE.     As  there  are  four  solutions  to  the  problem  of  drawing 
a  line  making  given  angles  with  the  principal  planes,  there  are 
also    four    solutions    to    this 
problem.      The   student   may 
ghow  these  cases  and  check  the 
accuracy  by  finding  the  angle 
between  a  given  plane  and  the 
principal  planes  (828). 

836.  Problem  26.  Through 
a  given  line,  in  a  given  plane, 
draw  another  line,  intersecting 
it  at  a  given  point,  and  at  a 
given  angle. 

Construction.  Let  AB, 
Fig.  141,  be  the  given  line,  T 
the  given  plane  and  G  the 
given  point.  Revolve  the 
limited  portion  of  the  plane 
tTt'  into  coincidence  with  the 

horizontal  plane.  Tt  remains  fixed  but  Tt'  revolves  to  Tt". 
To  find  the  direction  of  Tt",  consider  any  point  b'  on  the  original 
position  of  the  trace  Tt'.  The  distance  Tb'  must  equal  Tb"  as 


130 


GEOMETRICAL  PROBLEMS  IN   PROJECTION 


this  length  does  not  change  on  revolution;  since  Tt  is  the  axis  of 
revolution,  the  point  B  describes  a  circle,  the  plane  of  which  is 
perpendicular  to  the  axis,  and,  therefore,  b"  must  also  lie  on  a 
line  bb",  from  b,  perpendicular  to  the  trace  Tt. 

In  the  revolved  position  of  the  plane,  the  point  a  remains 
unchanged,  while  B  goes  to  b".  Hence,  ab"  is  the  revolved 
position  of  the  given  line  AB.  The  given  point  G  moves  to  g", 
on  a  line  gg",  perpendicular  to  Tt  and  must  also  be  on  the  line 
ab".  Through  g",  draw  the  line  cd",  making  the  required  angle 
a  with  it.  On  counter-revolution,  c  remains  fixed,  and  d" 
moves  to  d'.  Thus  CD  is  the  required  line,  making  an  angle 
a  with  another  line  AB,  and  lying  in  the  given  plane  T. 


FIG.  142. 

837.  Problem  27.  Through  a  given  line  in  a  given  plane, 
pass  another  plane,  making  a  given  angle  with  the  given  plane. 

Construction.  Let  T,  Fig.  142,  be  the  given  plane,  AB 
the  given  line  in  that  plane,  and  a  the  required  angle  between 
the  planes.  Construct  a  supplementary  view  of  the  line  AB; 
H'H'  is  the  new  horizontal  plane,  the  inclination  of  ab'  is  shown 
by  the  similar  letters  on  both  diagrams.  The  distance  of  b' 
above  the  horizontal  plane  must  also  equal  the  distance  b'  above 
H'H'  and  so  on.  Through  h,  in  the  supplementary  view,  draw 
a  plane  hf  perpendicular  to  ab'.  Revolve  h  about  f  to  g  and 
locate  g  as  shown  on  the  horizontal  projection  ab,  of  the  main 


THE  POINT,   THE   LINE,  AND  THE  PLANE 


131 


diagram.  The  line  eg  is  cut  from  the  plane  T,  by  the  auxiliary 
plane  egf,  hence,  lay  off  the  angle  a  as  shown.  This  gives 
the  direction  gf  of  the  line  cut  from  the  required  plane.  The 
point  f  lies  on  gf  and  also  on  ef  .which  is  perpendicular  to  ab. 
Hence,  join  af  and  produce  to  S;  join  S  and  b'  and  thus  estab- 


FIG.  143. 

lish  the  plane  S.,    The  plane  S  passes  through  the  given  line  AB 
and  makes  an  angle  a  with  the  given  plane  T  (827). 

838.  Problem  28.  To  construct  the  projections  of  a  circle 
lying  in  a  given  oblique  plane,  of  a  given  diameter,  its  centre 
in  the  given  plane  being  known. 

Construction.     Let  T,  Fig.  143,  be  the  given  plane,  and  G 


132 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


the  given  point  lying  in  the  plane  T.  When  g',  is  assumed,  for 
instance,  g  is  found  by  drawing  a  horizontal  line  g'a'  and  then 
ga  is  its  corresponding  projection;  g  can  therefore  be  determined 
as  shown.  When  the  plane  of  a  circle  is  inclined  to  a  plane  of 


FIG.  143. 

projection,  it  is  projected  as  an  ellipse.  An  ellipse  is  determined, 
and  can  be  constructed,  when  its  major  and  minor  axes  are  given.* 
To  construct  the  horizontal  projection,  revolve  the  plane  T 
about  Tt,  until  it  coincides  with  the  horizontal  plane  as  It". 
The  direction  of  Tt"  is  found  by  drawing  aa"  perpendicular  to 
Tt  and  laying  off  Ta'  =  Ta"  (836).  The  centre  of  the  circle  in 

*  For  methods  of  constructing  the  ellipse  see  Art.  906. 


THE  POINT,  THE  LINE,  AND  THE  PLANE  133 

its  revolved  position  is  found  by  drawing  a"g"  parallel  to  Tt 
and  gg"  perpendicular  to  Tt;  g"  is  this  revolved  position.  With 
the  given  radius,  draw  c"d"e"f",  the  circle  of  the  given  diameter. 
Join  e"d"  and  f"c";  prolong  these  lines  to  o  and  q,  and  also 
draw  g"p  parallel  to  these.  Thus,  three  parallel  lines  in  the 
revolved  position  of  the  plane  T,  are  established  and  on  counter 
revolution,  they  will  remain  parallel.  The  direction  gp,  of  one 
of  them  is  known,  hence,  make  fq  and  eo  parallel  to  gp.  The 
points  cdef  are  the  corresponding  positions  of  c"d"e"f"  and 
determine  the  horizontal  projection  of  the  circle.  The  line  ec 
remains  equal  to  e"c"  but  fd  is  shorter  than  f"d",  hence,  the 
major  and  minor  axes  of  an  ellipse  (projection  of  the  circle)  are 
determined.  The  ellipse  may  now  be  drawn  by  any  convenient 
method  and  the  horizontal  projection  of  the  circle  in  the  plane  T 
will  be  complete. 

By  revolving  the  plane  T  into  coincidence  with  the  vertical 
plane,  Tt'"  is  found  to  be  the  revolved  position  of  the  trace  Tt 
and  g"',  the  revolved  position  of  the  centre.  The  construction 
is  identical  with  the  construction  of  the  horizontal  projection 
and  will  become  apparent,  on  inspection,  as  the  necessary  lines 
are  shown  indicating  the  mode  of  procedure.  As  a  result,  k'l'm'n' 
determine  the  major  and  minor  axes  of  the  ellipse,  which  is  the 
vertical  projection  of  the  circle  in  the  plane  T. 

NOTE.  As  a  check  on  the  accuracy  of  the  work,  tangents 
may  be  drawn  in  one  projection  and  the  corresponding  projection 
must  be  tangent  at  the  corresponding  point  of  tangency. 


QUESTIONS  ON  CHAPTER  VIII 

1.  Mention  the  three  distinct  steps  into  which  the  solution  of  a  problem 

may  be  divided. 

2.  What  is  the  statement  of  a  problem? 

3.  What  is  the  analysis  of  a  problem? 

4.  What  is  the  construction  of  a,  problem? 

5.  What  type  of  projection  is  generally  used  in  the  construction  of  a 

problem? 

Note.  In  the  following  problems,  the  construction,  except  in  a  few 
isolated  cases,  is  to  be  entirely  limited  to  the  first  angle  of  pro- 
jection. 

6.  Draw  a  line  through  a  given  point,  parallel  to  a  given  line.     Give 

analysis  and  construction. 


134  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

7.  Draw  a  line  intersecting  a  given  line  at  a  given  point.    Give  analysis 

and  construction. 

8.  Find  where  a  given  oblique  line  pierces  the  planes  of  projection. 

Give  analysis  and  construction. 

9.  Transfer  the  diagram  of  Question  8  to  oblique  projection. 

10.  Pass  an  oblique  plane  through  a  given  oblique  line.     Give  analysis 

and  construction. 

11.  Make  an  oblique  projection  of  the  diagram  in  Question  10. 

12.  Pass  a  plane  obliquely  through  .the  principal  planes  and  through  a 

line  parallel  to  the  ground  line.     Give  analysis  and  construction. 
Hint:     Use  a  profile  plane  in  the  construction. 

13.  Transfer  the  diagram  of  Question  12  to  an  oblique  projection. 

14.  Pass  an  oblique  plane  through  a  line  which  is  parallel  to  the  hori- 

zontal plane  but  inclined  to  the  vertical  plane.     Give  analysis 
and  construction. 

15.  Transfer  the  diagram  of  Question  14  to  an  oblique  projection. 

16.  Pass  an  oblique  plane  through  a  line  which  is  parallel  to  the  vertical 

plane  but  inclined  to  the  horizontal  plane.     Give  analysis  and 
construction. 

17.  Transfer  the  diagram  of  Question  16  to  an  oblique  projection. 

18.  Pass  an  oblique  plane  through  a  given  point.     Give  analysis  and 

construction. 

19.  Make  an  oblique  projection  of  the  diagram  in  Question  18. 

20.  Find  the  intersection  of  two  planes,  oblique  to  each  other  and  to  the 

principal   planes.     Give   analysis   and   construction. 

21.  Transfer  the  diagram  of  Question  20  to  an  oblique  projection. 

22.  Find  the  intersection  of  two  planes,  oblique  to  each  other  and  to  the 

principal  planes.     Take  the  case  where  the  traces  do  not  intersect 
on  one  of  the  principal  planes.     Give  analysis  and  construction. 

23.  Find  the  corresponding  projection  of  a  given  point  lying  in  a  given 

oblique  plane,  when  one  projection  is  given.     Give  analysis  and 
construction. 

24.  Transfer  the  diagram  of  Question  23  to  an  oblique  projection. 

25.  Draw  a-  plane  which  contains  a  given  point  and  is  parallel  to  a  given 

plane.     Give  analysis  and  construction. 

26.  Transfer  the  diagram  of  Question  25  to  an  oblique  projection. 

27.  Draw  a  line  through  a  given  point,  perpendicular  to  a  given  plane. 

Give  analysis  and  construction. 

28.  Transfer  the  diagram  of  Question  27  to  an  oblique  projection. 

29.  Draw  a  plane  through  a  given  point,  perpendicular  to  a  given  line. 

Give  analysis  and  construction. 

30.  Transfer  the  diagram  of  Question  29  to  an  oblique  projection. 

31.  Pass  a  plane  through  three  given  points,  not  in  the  same  straight 

line.    Give  analysis  and  construction. 

32.  Transfer  the  diagram  of  Question  31  to  an  oblique  projection. 

33.  Revolve  a  given  point,  not  in  the  principal  planes,  about  a  line 

lying  in  one  of  the  principal  planes.    Give  analysis  and  construc- 
tion. 


THE  POINT,  THE  LINE,  AND  THE  PLANE  135 

34.  Transfer  the  diagram  of  Question  33  to  an  oblique  projection  and 

show  also  the  plane  of  the  revolving  point. 

35.  Find  the  true  distance  between  two  points  in  space,  when  both 

points  are  in  the  first  angle  of  projection.  Use  the  method  of 
revolving  the  projecting  plane  of  the  line  until  it  is  parallel  to  one 
plane  of  projection.  Give  analysis  and  construction. 

36.  Make  an  oblique  projection  of  the  diagram  in  Question  35  and  show 

the  projecting  plane  of  the  line. 

37.  Find  the  true  distance  between  two  points  in  space,  when  one  point 

is  in  the  first  angle  and  the  other  is  in  the  fourth  angle.  Use  the 
method  of  revolving  the  projecting  plane  of  the  line  until  it  is 
parallel  to  one  plane  of  projection.  Give  analysis  and  construction. 

38.  Make  an  oblique  projection  of  the  diagram  in  Question  37  and  show 

the  projecting  plane  of  the  line. 

39.  Find  the  true  distance  between  two  points  in  space,  when  both 

points  are  in  the  first  angle  of  projection.  Use  the  method  of 
revolving  the  projecting  plane  of  the  line  into  one  of  the  planes 
of  projection.  Give  analysis  and  construction. 

40.  Transfer  the  diagram  of  Question  39.  to  an   oblique  projection  and 

show  the  projecting  plane  of  the  line  and  the  path  described  by 
the  revolving  points. 

41.  Find  the  true  distance  between  two  points  in  space,  when  one  point 

is  in  the  first  angle  and  the  other  is  in  the  fourth  angle  of  pro- 
jection. Use  the  method  of  revolving  the  projecting  plane  of 
the  line  into  one  of  the  planes  of  projection.  Give  analysis  and 
construction. 

42.  Transfer  the  diagram  of  Question  39  to  an  oblique  projection  and 

show  the  projecting  plane  of  the  line  and  the  path  described  by 
the  revolving  points. 

43.  Find  where  a  given  line  pierces  a  given  plane.    Give  analysis  and 

construction. 

44.  Transfer  the  diagram  of  Question  43  to  an  oblique  projection. 

45.  Find  the  distance  of  a  given  point  from  a  given  plane.    Give  analysis 

and  construction. 

46.  Transfer  the  diagram  of  Question  45  to  an  oblique  projection.    Omit 

the  portion  of  the  construction  requiring  the  revolution  of  the 
points. 

47.  Find  the  distance  from  a  given  point  to  a  given  line.    Give  analysis 

and  construction. 

48.  Transfer  the  diagram  of  Question  47  to  an  oblique  projection.     Omit 

the  portion  of  the  construction  requiring  the  revolution  of  the 
points. 

49.  Find  the  angle  between  two  given  intersecting  lines.    Give  analysis 

and  construction. 

50.  Make  an  oblique  projection  of  the  diagram  in  Question  49. 

51.  Find  the  angle  between  two  given  planes.    Give  analysis  and  con- 

struction. 

52.  Make  an  oblique  projection  of  the  diagram  in  Question  51. 


136  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

53.  Find  the  angle  between  a  given  plane  and  the  horizontal  plane  of 

projection.     Give  analysis  and  construction. 

54.  Transfer  the  diagram  of  Question  53  to  an  oblique  projection. 

55.  Find  the  angle  between  a  given  plane  and  the  vertical  plane  of  pro- 

jection.    Give  analysis  and  construction. 

56.  Transfer  the  diagram  of  Question  55  to  an  oblique  projection. 

57.  Draw  a  plane  parallel  to  a  given  plane  at  a  given  distance  from  it. 

Give  analysis  and  construction. 

58.  Project  a  given  line  on  a  given  plane.     Give  analysis  and  construction. 

59.  Make  an  oblique  projection  of  the  diagram  in  Question  58. 

60.  Find  the  angle  between  a  given  line  and  a  given  plane.     Give  analysis 

and  construction. 

61.  Make  an  oblique  projection  of  the  diagram  in  Question  60. 

62.  Find  the   shortest   distance   between  a  pair  of  skew  lines.    Give 

analysis  and  construction. 

63.  Make  an  oblique  projection  of  the  diagram  in  Question  62. 

64.  Through  a  given  point,  draw  a  line  of  a  given  length,  making  given 

angles  with  the  planes  of  projection. 

65.  Show  the  construction  for  the  three  remaining  cases  of  the  problem 

in  Question  64. 

66.  Through  a  given  point,  draw  a  plane,  making  given  angles  with  the 

principal  planes. 

67.  Prove  that  the  construction  in  Question  66  is  correct  by  finding 

the  angles  that  the  given  plane  makes  with  the  principal  planes. 
(Note  that  there  are  four  possible  cases  of  this  problem.) 

68.  Show  the  construction  for  the  three  remaining  cases  of  the  problem 

in  Question  66. 

69.  Prove  that  the  constructions  in  Question  68  are  correct  by  finding 

the  angles  that  the  given  plane  makes  with  the  principal  planes. 

70.  Through  a  given  line,  in  a  given  plane,  draw  another  line  intersecting 

it  at  a  given  point,  and  at  a  given  angle. 

71.  Transfer  the  diagram  of  Question  70  to  an  oblique  projection. 

72.  Through  a  given  line  in  a  given  plane,  pass  another  plane,  making 

given  angles  with  the  given  plane. 

73.  Make  an  oblique  projection  of  the  diagram  in  Question  72. 

74.  Construct  the  projections  of  a  circle  lying  in  a  given  oblique  plane, 

the  diameter  and  its  centre  in  the  given  plane  being  known. 

Note.     The   following    exercises    embrace    operations    in   all   four 
angles. 

75.  Given  a  line  the  first  angle  and  a  point  in  the  second   angle,  draw 

a  line  through  the  given  point,  parallel  to  the  given  line. 

76.  Given  a  line  in  the  first  angle  and  a  point  in  the   third  angle,  draw 

a  line  through  the  given  point,  parallel  to  the  given  line. 

77.  Given  a  line  in  the  first  angle  and  a  point    in  the  fourth  angle,  draw 

a  line  through  the  given  point,  parallel  to  the  given  line. 

78.  Given  a  line  in    the   second  angle  and  a  point  in  the  third  angle, 

draw  a  line  through  the  given  point,  parallel  to  the  given  line. 


THE  POINT,  THE  LINE,  AND  THE  PLANE  137 

79.  Given  a  line  in  the  third  angle  and  a  point  in  the  fourth  angle,  draw 

a  line  through  the  given  point,  parallel  to  the  given  line. 

80.  Given  a  line  in  the  fourth  angle  and  a  point  in  the  second   angle, 

draw  a  line  through  the  given  point,  parallel  to  the  given  line. 

81.  Draw  two  intersecting  lines  in  the  second  angle  . 

82.  Draw  two  intersecting  lines  in  the  third  angle. 

83.  Draw  two  intersecting  lines  in  the  fourth  angle. 

84.  Find  where  a  given  line  pierces  the  principal  planes  when  the  limited 

portion  of  the  line  is  in  the  second  angle. 

85.  Find  where  a  given  line  pierces  the  principal  planes  when  the  limited 

portion  of  the  line  is  in  the  third  angle. 

86.  Find  where  a  given  line  pierces  the  principal  planes  when  the  limited 

portion  of  the  line  is  in  the  fourth  angle. 

87.  Show  the  second  angle  traces  of   a  plane  passed  through  a  second 

angle  oblique  line. 

88.  Show  the  third  angle  traces  of  a  plane  passed  through  a  third  angle 

oblique  line. 

89.  Show  the  fourth  angle  traces    of    a  plane  passed  through  a  fourth 

angle  oblique  line. 

90.  Given  a  line  in  the   second   angle  and  parallel  to  the  ground  line, 

pass  an  oblique  plane  through  it.  Use  a  profile  plane  as  part  of 
the  construction. 

91.  Given  a  line  in  the  third  angle  and  parallel  to  the    ground    line, 

pass  an  oblique  plane  through  it.     Use  a  profile  plane  as  a  part 
of  the  construction. 

92.  Given  a  line  in  the  fourth   angle    and  parallel  to  the  ground  lino, 

pass  an  oblique  plane  through  it.    Use  a  profile  plane  as  a  part 
of  the  construction. 

93.  Given  a  point   in  the  second  angle,  pass  an  oblique  plane  through 

it.     Show  only  second  angle  traces. 

94.  Given  a  point  in  the  third  angle,  pass  an  oblique  plane  through  it. 

Show  only  third  angle  traces. 

95.  Given  a  point  in  the  fourth  angle,  pass  an  oblique   plane  through 

it.     Show  only  fourth  angle  traces. 

96.  Given  the    second    angle  traces  of  two  oblique  planes,  find  their 

intersection  in  the  second  angle. 

97.  Given  the  third  angle  traces  of  two  oblique  planes,  find  their  inter- 

section in  the  third  angle. 

98.  Given  the  fourth  angle  traces  of  two    oblique    planes,  find  their 

intersection  in  the  fourth  angle. 

99.  Given  the  first  angle  traces  of  an  oblique  plane  and  one  projection 

of  a  second  angle  point,  find  the  corresponding  projection. 

100.  Given  the  first  angle  traces  of  an  oblique  plane,  and  one  projection 

of  a  third  angle  point,  find  the  corresponding  projection. 

101.  Given  the  first  angle  traces  of  an  oblique  plane,  and  one  projection 

of  a  fourth  angle  point,  find  the  corresponding  projection. 

102.  Given  the  second  angle  traces  of  an  oblique  plane,  and  one  projec- 

tion of  a  third  angle  point,  find  the  corresponding  projection. 


138  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

103.  Given  the  third  angle  traces  of  an  oblique  plane,  and  one  pro- 

jection of  a  fourth  angle  point,  find  the  corresponding  pro- 
jection. 

104.  Given  the  first  angle  traces  of  an  oblique  plane,  and  a  point  in  the 

second  angle,  pass  a  plane  through  the  given  point  and  parallel 
to  the  given  plane. 

105.  Given  the  first  angle  traces  of  an  oblique  plane,  and  a  point  in  the 

third  angle,  pass  a  plane  through  the  given  point  and  parallel  to 
the  given  plane. 

106.  Given  the  first  angle  traces  of  an  oblique  plane,  and  a  point  in  the 

fourth  angle,  pass  a  plane  through  the  given  point  and  parallel 
to  the  given  plane. 

107.  Given  the  second  angle  traces  of  an  oblique  plane,  and  a  point  in 

the  third  angle,  pass  a  plane  through  the  given  point  and  parallel 
to  the  given  plane. 

108.  Given  the  third  angle  traces  of  an  oblique  plane,  and  a  point  in 

the  fourth  angle,  pass  a  plane  through  the  given  point  and  parallel 
to  the  given  plane. 

109.  Given  the  first  angle  traces  of  an  oblique  plane  and  a  point  in  the 

second  angle,  draw  a  line  through  the  given  point  perpendicular 
to  the  given  plane. 

110.  Given  the  first  angle  traces  of  an  oblique  plane  and  a  point  in  the 

third  angle,  draw  a  line  through  the  given  point  perpendicular 
to  the  given  plane. 

111.  Given  the  first  angle  traces  of  an  oblique  plane  and  a  point  in  the 

fourth  angle,  draw  a  line  through  the  given  point  perpendicular 
to  the  given  plane. 

112.  Given  the  second  angle  traces  of  an  'oblique  plane  and  a  point  in 

the  fourth  angle,  draw  a  line  through  the  given  point  perpendicular 
to  the  given  plane. 

113.  Given  the  first  angle  traces  of  a  plane  perpendicular  to  the  hori- 

zontal plane  but  inclined  to  the  vertical  plane,  and  a  point  in 
the  third  angle,  draw  a  line  through  the  given  point  perpendicular 
to  the  given  plane. 

114.  Given  the  third  angle  traces  of  a  plane  perpendicular  to  the  vertical 

plane  but  inclined  to  the  horizontal  plane,  and  a  point  in  the 
fourth  angle,  draw  a  line  through  the  given  point,  perpendicular 
to  the  given  plane. 

115.  Given  the  first  angle  traces  of  a  plane  parallel  to  the  ground  line 

and  a  point  in  the  third  angle,  draw  a  line  through  the  given  point 
perpendicular  to  the  given  plane. 

116.  Draw  a  profile  plane  of  the  diagram  in  Question  115. 

117.  Make  an  oblique  projection  of  the  diagram  in  Question  115. 

118.  Given  the  fourth  angle  traces  of  a  plane  parallel  to  the  ground  line 

and  a  point  in  the  second  angle,  draw  a  line  through  the  given 
point  perpendicular  to  the  given  plane. 

119.  Draw  a  profile  plane  of  the  diagram  in  Question  118. 

120.  Make  an  oblique  projection  of  the  diagram  in  Question  118. 


THE   POINT,   THE  LINE,  AND  THE  PLANE  139 

121.  Given  a  line  in  the  first  angle  and  a  point  in  the  second  angle,  pass 

a    plane  through  the  given  point  perpendicular  to  the  given 
line. 

122.  Given  a  line  in  the  first  angle  and  a  point  in  the  third  angle,  pass  a 

plane  through  the  given  point  perpendicular  to  the  given  line. 

123.  Given  a  line  in  the  first  angle  and  a  point  in  the  fourth  angle,  pass 

a  plane  through  the  given  point  perpendicular  to  the  given  line. 

124.  Given  a  line  in  the  second  angle  and  a  point  in  the  third  angle, 

pass  a  plane  through  the  given  point  perpendicular  to  the  given 
line. 

125.  Given  a  line  in  the  third  angle  and  a  point  in  the  fourth  angle,  pass 

a  plane  through  the  given  point  perpendicular  to  the  given  line. 

126.  Given  three  points  in  the  second  angle,  pass  a  plane  through  them. 

127.  Given  three  points  in  the  third  angle,  pass  a  plane  through  them. 

128.  Given  three  points  in  the  fourth  angle,  pass  a  plane  through  them. 

129.  Given  two  points  in  the  first  angle  and  one  point  in  the  second 

angle,  pass  a  plane  through  them. 

130.  Given  a  point  in  the  first  angle,  one  in  the  second  angle,  and  one 

in  the  third  angle,  pass  a  plane  through  them. 

131.  Given  a  point  in  the  second  angle,  one  in  the  third  angle,  and  one 

in  the  fourth  angle,  pass  a  plane  through  them. 

132.  Given  a  line  in  the  horizontal  plane  and  a  point  in  the  second  angle, 

revolve  the  point  about  the  line  until  it  coincides  with  the  hori- 
zontal plane. 

133.  Given  a  line  in  the  horizontal  plane  and  a  point  in  the  third  angle, 

revolve  the  point  about  the  line  until  it  coincides  with  the  hori- 
zontal plane. 

134.  Given  a  line  in  the  horizontal  plane  and  a  point  in  the  fourth  angle 

revolve  the  point  about  the  line  until  it  coincides  with  the  hori- 
zontal plane. 

135.  Given  a  line  in  the  vertical  plane  and  a  point  in  the  second  angle, 

revolve  the  point  about  the  line  until  it  coincides  with  the  vertical 
plane. 

136.  Given  a  line  in  the  vertical  plane  and  a  point  in  the  fourth  angle, 

revolve  the  point  about  the  line  until  it  coincides  with  the  vertical 
plane. 

137.  Given  two  points  in  the  second  angle,  find  the  true  distance  between 

them. 

138.  Given  two  points  in  the  third  angle,  find  the  true  distance  between 

them. 

139.  Given  two  points  in  the  fourth  angle,  find  the  true  distance  between 

them. 

140.  Given  one  point  in  the  first  angle  and  one  point  in  the  second  angle, 

find  the  true  distance  between  them. 

141.  Given  one  point  in  the  first  angle  and  one  point  in  the  third  angle, 

find  the  true  distance  between  them. 

142.  Given  one  point  in  the  second  angle  and  one  point  in  the  third 

angle,  find  the  true  distance  between  them. 


140  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

143.  Given  one  point  in  the  second  angle  and  one  point  in  the  fourth 

angle,  find  the  true  distance  between  them. 

144.  Given  the  first  angle  traces  of  a  plane  and  a  line  in  the  second  angle, 

find  where  the  given  line  pierces  the  given  plane. 

145.  Given  the  first  angle  traces  of  a  plane  and  a  line  in  the  third  angle, 

find  where  the  given  line  pierces  the  given  plane. 

146.  Given  the  first  angle  traces  of  a  plane  and  a  line  in  the  fourth  angle, 

find  where  the  given  line  pierces  the  given  plane. 

147.  Given  the  second  angle  traces  of  a  plane  and  a  line  in  the  third 

angle,  find  where  the  given  line  pierces  the  given  plane. 

148.  Given  the  second  angle  traces  of  a  plane  and  a  line  in  the  fourth 

angle,  find  where  the  given  line  pierces  the  given  plane. 

149.  Given  the  third  angle  traces  of  a  plane  and  a  line  in  the  fourth 

angle,  find  where  the  given  line  pierces  the  given  plane. 

150.  Given  the  first  angle  traces  of  a  plane  and  a  point  in  the  second 

angle,  find  the  distance  from  the  point  to  the  plane. 

151.  Given  the  first  angle  traces  of  a  plane  and  a  point  in  the  third  angle, 

find  the  distance  from  the  point  to  the  plane. 

152.  Given  the  first  angle  traces  of  a  plane  and  a  point  in  the  fourth 

angle,  find  the  distance  from  the  point  to  the  plane. 

153.  Given  the  second  angle  traces  of  a  plane  and  a  point  in  the  third 

angle,  find  the  distance  from  the  point  to  the  plane. 

154.  Given  the  second  angle  traces  of  a  plane  and  a  point  in  the  fourth 
*     angle,  find  the  distance  form  the  point  to  the  plane. 

155.  Given  the  third  angle  traces  of   a  plane  and  a  point  in  the  fourth 

angle,  find  the  distance  from  the  point  to  the  plane. 

156.  Given  a  line  in  the  first  angle  and  a  point  in  the  second  angle,  find 

the  distance  from  the  point  to  the  line. 

157.  Given  a  line  in  the  first  angle  and  a  point  in  the  third  angle,  find 

the  distance  from  the  point  to  the  line. 

158.  Given  a  line  in  the  first  angle  and  a  point  in  the  fourth  angle,  find 

the  distance  from  the  point  to  the  line. 

159.  Given  a  line  in  the  second  angle  and  a  point  in  the  third  angle, 

find  the  distance  from  the  point  to  the  line. 

160.  Given  a  line  in  the  second  angle  and  a  point  in  the  fourth  angle, 

find  the  distance  from  the  point  to  the  line. 

161.  Given  a  line  in  the  third  angle  and  a  point  in  the  fourth  angle,  find 

"  the  distance  from  the  point  to  the  line. 

162.  Given  two  intersecting  lines  in  the  second  angle,  find  the  angle 

between  them. 

163.  Given  two  intersecting  lines  hi  the  third  angle,  find  the  angle  between 

them. 

164.  Given  two  intersecting  lines  in  the  fourth  angle,  find  the  angle 

between  them. 

165.  Given  two  intersecting  lines,  one  in  the  first  and  one  in  the  second 

angle,  find  the  angle  between  them. 

166.  Given  two  intersecting  lines,  one  in  the  first  and  one  in  the  third 

angle,  find  the  angle  between  them. 


THE  POINT,   THE  LINE,   AND  THE   PLANE  141 

167.  Given  two  intersecting  lines,  one  in  the  first  and  one  in  the  fourth 

angle,  find  the  angle  between  them. 

168.  Given  two  intersecting  lines,  one  in  the  second  and  one  in  the  third 

angle,  find  the  angle  between  them. 

169.  Given  two  intersecting  lines,  one  in  the  second  and  one  in  the  fourth 

angle,  find  the  angle  between  them. 

170.  Given  two  intersecting  lines,  one  in  the  third  and  one  in  the  fourth 

angle,  find  the  angle  between  them. 

171.  Given  the  second  angle  traces  of  two  intersecting  planes,  find  the 

angle  between  them. 

172.  Given  the  third  angle  traces  of  two  intersecting  planes,  find  the 

angle  between  them. 

173.  Given  the  fourth  angle  traces  of  two  intersecting  planes,  find  the 

angle  between  them. 

174.  Given  the  first  angle  traces  of  one  plane  and  the  second  angle  traces 

of  another  plane,  find  the  angle  between  them. 

175.  Given  the  first  angle  traces  of  one  plane  and  the  third  angle  traces 

of  another  plane,  find  the  angle  between  them. 

176.  Given  the  first  angle  traces  of  one  plane  and  the  fourth  angle  traces 

of  another  plane,  find  the  angle  between  them. 

177.  Given  the  second  angle  traces  of  one  plane  and  the  third  angle 

traces  of  another  plane,  find  the  angle  between  them. 

178.  Given  the  second  angle  traces  of  one  plane  and  the  fourth  angle 

traces  of  another  plane,  find  the  angle  between  them. 

179.  Given  the  third  angle  traces  of  one  plane  and  the  fourth  angle  traces 

of  another  plane,  find  the  angle  between  them. 

180.  Given  a  plane  hi  the  second  angle,  find  the  angle  between  the  given 

plane  and  the  horizontal  plane. 

181.  Given  a  plane  in  the  second  angle,  find  the  angle  between  the  given 

plane  and  the  vertical  plane. 

182.  Given  a  plane  in  the  third  angle,  find  the  angle  between  the  given 

plane  and  the  horizontal  plane. 

183.  Given  a  plane  in  the  third  angle,  find  the  angle  between  the  given 

plane  and  the  vertical  plane. 

184.  Given  a  plane  in  the  fourth  angle,  find  the  angle  between  the  given 

plane  and  the  horizontal  plane. 

185.  Given  a  plane  in  the  fourth  angle,  find  the  angle  between  the  given 

plane  and  the  vertical  plane. 

186.  Given  the  second  angle  traces  of  a  plane,  draw  another  parallel 

plane  at  a  given  distance  from  it. 

187.  Given  the  third  angle  traces  of  a  plane,  draw  another  parallel  plane 

at  a  given  distance  from  it. 

188.  Given  the  fourth  angle  traces  of  a  plane,  draw  another  parallel 

plane  at  a  given  distance  from  it. 

189.  Given  the  first  angle  traces  of  a  plane  and  a  line  in  the  second  angle, 

project  the  given  line  on  the  given  plane. 

190.  Given  the  first  angle  traces  of  a  plane  and  a  line  in  the  third  angle, 

project  the  given  line  on  the  given  plane. 


142  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

191.  Given  the  first  angle  traces  of  a  plane  and  a  line  in  the  fourth  angle, 

project  the  given  line  on  the  given  plane. 

192.  Given  the  second  angle  traces  of  a  plane  and  a  line  in  the  third  angle, 

project  the  given  line  on  the  given  plane. 

193.  Given  the  second  angle  traces  of  a  plane  and  a  line  in  the  fourth 

angle,  project  the  given  line  on  the  given  plane. 

194.  Given  the  third  angle  traces  of  a  plane  and  a  line  in  the  fourth  angle, 

project  the  given  line  on  the  given  plane. 

195.  Given  the  fourth  angle  traces  of  a  plane  and  a  line  in  the  second 

angle,  project  the  given  line  on  the  given  plane. 

196.  Given  the  first  angle  traces  of  a  plane  and  a  line  in  the  second  angle, 

find  the  angle  between  the  given  line  and  the  given  plane. 

197.  Given  the  first  angle  traces  of  a  plane  and  a  line  in  the  third  angle, 

find  the  angle  between  the  given  line  and  the  given  plane. 

198.  Given  the  first  angle  traces  of  a  plane  and  a  line  in  the  fourth  angle, 

find  the  angle  between  the  given  line  and  the  given  plane. 

199.  Given  the  second  angle  traces  of  a  plane  and  a  line  in  the  third  angle, 

find  the  angle  between  the  given  line  and  the  given  plane. 

200.  Given  the  second  angle  traces  of  a  plane  and  a  line  in  the  fourth 

angle,  find  the  angle  between  the  given  line  and  the  given  plane 

201.  Given  the  third  angle  traces  of  a  plane  and  a  line  in  the  fourth  angle, 

find  the  angle  between  the  given  line  and  the  given  plane. 

202.  Given  the  fourth  angle  traces  of  a  plane  and  a  line  in  the  second 

angle,  find  the  angle  between  the  given  line  and  the  given  plane. 

203.  Given  the  fourth  angle  traces  of  a  plane  and  a  line  in  the  third  angle, 

find  the  angle  between  the  given  line  and  the  given  plane. 

204.  Given  a  line  in  the  first  angle  and  another  line  in  the  second  angle 

(skew  lines),  find  the  shortest  distance  between  them. 

205.  Given  a  line  in  the  first  angle  and  another  line  in  the  third  angle 

(skew  lines),  find  the  shortest  distance  between  them. 

206.  Given  a  line  in  the  first  angle  and  another  line  in  the  fourth  angle 

(skew  lines),  find  the  shortest  distance  between  them. 

207.  Given  a  line  in  the  second  angle  and  another  line  in  the  third  angle 

(skew  lines),  find  the  shortest  ditstance  between  them. 

208.  Given  a  line  in  the  second  angle  and  another  line  in  the  fourth 

angle  (skew  lines) ;  find  the  shortest  distance  between  them. 

209.  Given  a  line  in  the  third  angle  and  another  line  in  the  fourth  angle 

(skew  lines),  find  the  shortest  distance  between  them. 

210.  Through  a  given  point  in  the  second  angle,  draw  a  line  of  a  given 

length,  making  given  angles  with  the  planes  of  projection. 

211.  Through  a  given  point  in  the  third  angle,  draw  a  line  of  a  given 

length,  making  given  angles  with  the  planes  of  projection. 

212.  Through  a  given  point  in  the  fourth  angle,  draw  a  line  of  a  given 

length,  making  given  angles  with  the  planes  of  projection. 

213.  Through  a  given  point  in  the  second  angle,  draw  a  plane  making 

given  angles  with  the  principal  planes. 

214.  Through  a  given  point  in  the  third  angle,  draw  a  plane  making 

given  angles  with  the  principal  planes. 


THE  POINT,  THE  LINE,   AND  THE  PLANE  143 

215.  Through  a  given  point  in  the  fourth  angle,  draw  a  plane  making 

given  angles  with  the  principal  planes. 

216.  Given  the  second  angle  traces  of  a  plane,  a  line,  and  a  point  in  the 

plane,  draw  another  line  through  the  given  point,   making  a 
given  angle  with  the  given  line. 

217.  Given  the  third  angle  traces  of  a  plane,  a  line,  and  a  point  in  the 

plane,   draw  another  line  through  the  given   point,   making  a 
given  angle  with  the  given  line. 

218.  Given  the  fourth  angle  traces  of  a  plane,  a  line  and  a  point  in  the 

plane,  draw  another  line  through  the  given  point,  making  a  given 
angle  with  the  given  line. 

219.  Given  the  second  angle  traces  of  a  plane,  and  a  line  in  the  plane, 

draw  through  the  line  another  plane  making  a  given  angle  with 
the  given  plane. 

220.  Given  the  third  angle  traces  of  a  plane,  and  a  line  in  the  plane, 

draw  through  the  line  another  plane  making  a  given  angle  with 
the  given  plane. 

221.  Given  the  fourth  angle  traces  of  a  plane,  and  a  line  in  the  plane, 

draw  through  the  line  another  plane  making  a  given  angle  with 
the  given  plane. 

222.  Given  the  second  angle  traces  of  a  plane,  the  diameter  and  the 

centre  of  a  circle,  construct  the  projections  of  the  circle. 

223.  Given  the  third  angle  traces  of  a  plane,  the  diameter  and  the  centre 

of  a  circle,  construct  the  projections  of  the  circle. 

224.  Given  the  fourth  angle  traces  of  a  plane,  the  diameter,  and  the 

centre  of  a  circle,  construct  the  projections  of  the  circle, 


CHAPTER  IX 
CLASSIFICATION  OF  LINES 

901.  Introductory.     Lines  are  of  an  infinite  variety  of  forms. 
The  frequent  occurrence  in  engineering  of  certain  varieties  makes 
it  desirable  to  know  their  properties  as  well  as  their  method 
of  construction.     It  must  be  remembered  that  lines  and  points 
are   mathematical   concepts   and   that   they   have   no   material 
existence.     That  is  to  say,  a  line  may  have  so  many  feet  of  length 
but  as  it  has  no  width  or  thickness,  its  volume,  therefore,  is 
zero.     Hence,  it  cannot  exist  except  in  the  imagination.     Like- 
wise, a  point  is  a  still  further  reduction  and  has  position  only; 
it  has  no  dimensions  at  all.     Of  course,  the  rraterial  representa- 
tion of  lines  and  points  requires  finite  dimensions,   but  when 
speaking  of  them,   or  representing  them,   it  is  the  associated 
idea,  rather  than  the  representation,  which  is  desired. 

902.  Straight  line.    A  straight  line  may  be  defined  as  the 
shortest  distance  between  two  points.*     It  may  also  be  described 
as  the  locus  (or  path)  of  a  generating  point  which  moves  in  the 
same  direction.     Hence,  a  straight  line  is  fixed  in  space  by  two 
points,  or,  by  a  point  and  a  direction. 

903.  Singly  curved  line.    A  singly  curved  line  is  the  locus 

(plural  :-loci)  of  a  generating  point  which  moves  in  a  varying 
direction  but  remains  in  a  single  plane.  Sometimes  the  singly 
curved  line  is  called  a  plane  curve  because  all  points  on  the 
curve  must  lie  in  the  same  plane. 

904.  Representation  of  straight  and  singly  curved  lines. 

Straight  and  singly  curved  lines  are  represented  by  their  pro- 
jections. When  singly  curved  lines  are  parallel  to  the  plane 

*  Frequently,  this  is  called  a  right  line.     There  seems  no  reason,  however, 
why  this  new  nomenclature  should  be  used;  hence,  it  is  here  avoided. 

144 


CLASSIFICATION  OF  LINES  145 

of  projection,  they  are  projected  in  their  true  form  and  require 
only  one  plane  of  projection  for  their  complete  representation. 
If  the  plane  of  the  curve  is  perpendicular  to  the  principal  planes, 
a  profile  will  suffice.  If  the  plane  of  the  curve  is  inclined  to  the 
planes  of  projection,  both  horizontal  and  vertical  projections 
may  be  necessary,  unless  a  supplementary  plane  be  used,  which 
is  parallel  to  the  plane  of  the  curve.*  This  latter  condition  is 
then  similar  to  that  obtained  wyhen  the  plane  of  the  curve  is 
parallel  to  one  of  the  principal  planes. 

905.  Circle.  The  circle  is  a  plane  curve,  every  point  of  which 
is  equidistant  from  a  fixed  point  called  the  centre.  The  path 
described  -by  the  moving  point  (or  locus)  is  frequently  designated 
as  the  " circumference  of  the  circle."!  To  define  such  curves 
consistently,  it  is  necessary  to  limit  the  definition  to  the  nature 
of  the  line  forming  the  curve.  Therefore,  what  is  usually  known 
as  the  circumference  of  the  circle  should  simply  be  known  as 
the  circle  and  then  subsequent  definitions  of  other  curves  become 
consistently  alike. 

In  Fig.  144  a  circle  is  shown;  abedc  is  the  curve  or  circle 
proper.  The  fixed  point  o  is  called  the  centre 
and  every  point  on  the  circle  is  equidistant  from 
it.  This  fixed  distance  is  called  the  radius ;  oa, 
oc,  and  ob  are  all  radii  of  the  circle;  be  is  a 
diameter  and  is  a  straight  line  through  the 
centre  and  equal  to  two  radii  in  length.  The 
straight  line,  de,  limited  by  the  circle  is  a  chord 
and  when  it  passes  through  the  centre  it 
becomes  a  diameter;  when  the  same  line  is  extended  like  gh  it  is 
a  secant.  A  limited  portion  of  the  circle  like  ac  is  an  arc;  when 
equal  to  one-quarter  of  the  whole  circle  it  is  a  quadrant;  when 
equal  to  one-half  of  the  whole  circle  it  is  a  semicircle  as  cab 
or  bfc.  The  area  included  between  two  radii  and  the  circle  is 
a  sector  as  aoc;  that  between  any  line  like  de  and  the  circle  is 

*  In  all  these  cases,  the  plane  of  the  curve  may  be  made  to  coincide  with 
the  plane  of  projection.  This,  of  course,  is  a  special  case. 

t  The  introduction  of  this  term  makes  it  necessary  to  define  the  circle 
as  the  enclosed  area.  This  condition  is  unfortunate,  as  in  the  case  of  the 
parabola,  hyperbola,  and  numerous  other  curves,  the  curves  are  open  and  do 
not  enclose  an  area. 


146 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


a  segment.  If  any  other  circle  is  drawn  with  the  same  centre  of 
the  circles  are  concentric,  otherwise  eccentric;  the  distance 
between  the  centres  in  the  latter  case  is  the  eccentricity. 

906.  Ellipse.  The  ellipse  is  a  plane  curve,  in  which  the 
sum  of  the  distances  of  any  point  on  the  curve  from  two  fixed 
points  is  constant.  Fig.  145  shows  this  curve  as  adbc;  The 
major  axis  is  ab,  and  its  length  is  the  constant  distance  of  the 
definition;  cd  is  the  minor  axis  and  is  perpendicular  to  ab  through 
its  middle  point  o.  The  fixed  points  or  foci  (either  one  is  a  focus) 
are  e  and  f  and  are  located  on  the  major  axis. 

If  ab  be  assumed  as  the  constant  distance,  and  e  and  f  be 
assumed  as  the  foci,  the  minor  axis  is  determined  by  drawing 
arcs  from  e  and  f  with  oa,  equal  to  one-half  of  the  major  axis, 


FIG.  145. 


FIG.  146. 


as  a  radius.  Thus  ec+cf  =  ab  and  also  ed+df  =  ab.  To  find 
any  other  point  on  the  curve,  assume  any  distance  as  bg,  and 
with  bg  as  a  radius  and  f  as  a  centre,  draw  an  arc  fh.  With 
the  balance  of  the  major  axis  ag  as  a  radius,  draw  an  arc  eh 
from  the  focus  e  as  a  centre.  These  intersecting  arcs  locate  h, 
a  point  on  the  curve.  Thus  eh-hhf  =  ab  and  therefore  satisfies 
the  definition  of  the  curve.  With  the  same  radii  just  used,  the 
three  other  points  i,  j  and  k  are  located.  In  general,  four  points 
are  determined  for  any  assumption  except  for  the  points  a,  c,  b 
or  d.  It  can  be  shown  that  a  and  b  are  points  on  the  curve, 
because  oa  =  ob  and  oe  =  of.  Hence  ae=fb,  therefore,  fa+ea  = 
ab  and  also  eb+fb  =  ab. 

In  the  construction  of  this,  or  any  other  curve,  the  student 
should  avoid  trying  to  save  time  by  locating  only  a  few  points. 
This  is  a  mistaken  idea,  as,  within  reasonable  limits,  time  is 
saved  by  drawing  numerous  points  on  the  curve,  particularly 


CLASSIFICATION  OF  LINES 


147 


where  the  curve  changes  its  direction  rapidly.  The  direction  of 
the  curve  at  any  point  should  be  known  with  a  reasonable  degree 
of  accuracy. 

Another  method  of  drawing  an  ellipse  is  shown  in  Fig.  146. 
It  is  known  as  the  trammel  method.  Take  any  straight  ruler 
and  make  oc  =  a  and  od=b.  By  locating  c  on  the  minor  axis 
and  d  on  the  major  axis,  a  point  is  located  at  o,  as  shown.  This 
method  is  a  very  rapid  one,  and  is  the  one  generally  used  when 
a  true  ellipse  is  to  be  plotted,  on  account  of  the  very  few  lines 
required  in  the  construction.  Both  methods  mentioned  are 
theoretically  accurate,  but  the  latter  method  is  perhaps  used 
oftener  than  the  former. 

In  practice,  ellipses  are  usually  approximated  by  employing 
four  circular  arcs,  of  two  different  radii  as  indicated  in  Art.  405. 
The  major  and  minor  axes  are  laid  off  and  a  smooth  looking 
curve  drawn  between  these  limits.  Of  course,  the  circular  arcs 
do  not  produce  a  true  ellipse,  but  as  a  rule,  this  method  is  a 
rapid  one  and  answers  the  purpose  in  conveying  the  idea. 

907.  Parabola.    The  parabola  is  a  plane  curve,  which  is 
the  locus  of  a  point,  moving  so  that  the  distance  from  a  fixed 
point    is    always    equal    to    the    distance 

from  a  fixed  line.  The  fixed  point  is  the 
focus  and  the  fixed  line  is  the  directrix. 
A  line  through  the  focus  and  perpen- 
dicular to  the  directrix  as  eg,  Fig.  147, 
is  the  axis  with  respect  to  which  the  curve 
is  symmetrical.  The  intersection  of  the 
axis  and  the  curve  is  the  vertex  shown 
at  f . 

The  point  a  on  the  curve  is  so  situated 
that  ac  =  ad;  also  b  is  so  situated  that 
be  =  be.  Points  beyond  b  become  more  and 

more  remote,  from  both  the  axis  and  directrix.  Hence,  it  is' 
an  open  curve,  extending  to  infinity.  Discussions  are  usually 
limited  to  some  finite  portion  of  the  curve. 

908.  Hyperbola.    The  hyperbola  is  a  plane  curve,  traced 
by  a  point,  which  moves  so  that  the  difference  of  its  distances 
from  two  fixed  points  is  constant.  The  fixed  points  a  and  b, 


FIG.  147. 


148 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


FIG.  148. 


Fig.  148,  are  the  foci.     The  line  ab,  passing  through  them,  is 
the  transverse  *  axis ;  the  point  at  which  either  curve  crosses 
the  axis  as  e  or  f  is  the  vertex  (plural: — vertices),  and  the  line  kl, 
perpendicular  to  ab  at  its  middle  point,  is  the  conjugate  axis. 
To  draw  the  curve,  lay  off  the  foci  a  and  b;  also  lay  off  ef, 

the  constant  distance,  so  that 
eo  =  of  and  o  is  the  middle 
of  ab.  It  must  be  observed 
that  ef  is  always  smaller  than 
ab  otherwise  the  curve  cannot 
be  constructed.  Take  any 
radius  be,  greater  than  bf, 
and  draw  an  indefinite  arc; 
from  a,  draw  ac,  so  that  ac 
—  bc  =  ef,  hence,  ac=bc-f-ef. 
The  point  c  is  thus  on  the 
curve.  Similarly,  draw  an 
arc  ad,  from  a,  and  another 
arc  bd  =  ad + ef .  This  locates 
d,  which,  in  this  case,  is  on 

another  curve.  In  spection  will  show  that  there  are  two 
branches  of  this  curve.  The  point  c  has  been  selected  so  as  to 
be  on  one  branch,  and  d,  on  the  other.  To  construct  the  curve, 
accurately,  many  more  points  must  be  located  than  shown. 
Both  branches  are  open  and  symmetrical  with  respect  to  both 
axes. 

A  tangent  to  the  curve  through  o  is  called  an  asymptote 
when  it  touches  the  hyperbola  in  two  points,  each  at  an  infinite 
distance  from  o.  As  will  be  observed,  there  are  two  asymptotes. 

909.  Cycloid.  The  cycloid  is  a  plane  curve,  traced  by  a 
point  on  a  circle  which  rolls  over  a  straight  line.  The  straight 
line  over  which  the  circle  rolls  is  the  directrix;  the  point  on  the 
circle  may  be  considered  as  the  generating  point. 

The  curve  is  shown  as  abc  . .  .  i  in  Fig.  149.  To  construct 
it,  lay  out  the  directrix  ai,  and  to  one  side,  draw  an  auxiliary 
circle  equal  in  diameter  to  the  rolling  circle  as  shown  at  1'  2', 
etc.  On  a  line  through  the  centre  of  the  auxiliary  circle,  draw 
the  line  3'-9  parallel  to  the  directrix  ai.  Lay  off  the  distance 

*  Sometimes  known  as  the  "principal  axis." 


CLASSIFICATION  OF  LINES 


149 


1-9  on  this  line,  equal  to  the  length  of  the  circle  (3. 14 X diameter). 
Assume  that  when  the  centre  of  the  rolling  circle  is  at  1,  a  is 
the  position  of  the  generating  point.  After  one-eighth  of  a 
revolution  the  centre  has  moved  to  2,  where  the  distance  1-2 
is  one-eighth  of  the  distance  1-9.  The  corresponding  position 
of  the  generating  point  is  shown  at  2'  in  the  auxiliary  circle  at 
the  left.  Hence,  for  one-eighth  of  a  revolution  the  centre  of  the 
rolling  circle  has  moved  from  1  to  2  and  the  generating  point 
has  moved  a  distance  vertically  upward  equal  to  the  distance 
of  2'  above  the  line  ai.  Therefore,  draw  a  line  through  2',  parallel 
to  ai;  and  then  from  2  as  a  centre,  draw  an  arc  with  the  radius 
of  the  rolling  circle  intersecting  this  line  in  the  point  b,  a  point 
on  the  required  curve.  After  a  quarter  of  a  revolution,  the  gen- 
erating point  is  above  the  directrix,  at  a  height  equal  to  the 


FIG.  149. 


distance  of  3'  above  ai.  It  is  also  on  an  arc  from  3,  with  a  radius 
equal  again  to  the  radius  of  the  rolling. circle;  hence,  c  is  the  point. 
This  process  is  continued  until  the  generating  point  reaches  its 
maximum  height  e  after  one-half  of  a  revolution,  when  it  begins 
to  descend,  to  i,  as  shown,  after  completing  one  revolution. 
Further  rolling  of  the  circle  causes  the  generating  point  to  dupli- 
cate its  former  steps,  as  it  continues  along  the  directrix  to  infinity 
if  desired. 

The  cycloid  is  the  same  curve  that  is  produced  by  a  mark 
on  the  rim  of  a  car-wheel,  while  rolling  along  the  track,  only, 
here,  no  slipping  is  permitted.  The  cycloid  is,  thus,  continuous, 
each  arch  being  an  exact  duplicate  of  the  preceding. 

910.  Epicycloid.  The  epicyloid  is  a  plane  curve,  which  is 
generated  by  a  point  on  a  circle  which  rolls  on  the  outside  of 
another  circle.  The  directrix  is  here  an  arc  of  a  circle,  instead 
of  a  straight  line. 


150 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


The  construction  is  indicated  in  Fig.  150.  The  centre  of  the 
rolling  circle  assumes  successive  positions  as  1,  2,  3,  etc.  The 
length  of  the  arc  ai  is  equal  to  the  length  of  the  rolling  circle. 
The  points  are  located  much  the  same  as  in  the  cycloid,  and,  as 


the  necessary  construction  lines  are  shown,  the  student  should 
have  no  difficulty  in  following  the  construction. 

911.  Hypocycloid.    The    hypocycloid    is    a    plane    curve/ 
generated  by  a  point  on  a  circle  which  rolls  on  the  inside  o 


FIG.  151. 


another  circle.  The  directrix  of  the  epicycloid  and  the  hypocy- 
cloid may  be  the  same,  the  epicycloid  is  described  on  the  outside 
of  the  arc  while  the  hypocycloid  is  described  on  the  inside. 

All  the  necessary  construction  lines  required  to  draw  the 
hypocycloid  have  been  added  in  Fig.   151  and  the  method  is 


CLASSIFICATION  OF  LINES  151 

perhaps  evident.  No  confusion  should  arise  even  though  the 
curves  are  drawn  in  positions  other  than  those  shown.  For 
instance,  the  cycloids  (this  includes  the  cycloids,  epicycloid  and 
hypocycloid)  could  be  shown  upside-down;  the  curves  are  gener- 
ated in  much  the  same  way  and  their  properties,  therefore,  do 
not  change. 

912.  Spiral.     The  spiral  is  a  plane  curve,  generated  by  a 
point  moving  along  a  given  line  while  the  given  line  is  revolving 
about  some  point  on  the  line.     An  infinite  number  of  spirals 
may  exist,  because  the  point  may  have  a  variable  velocity  along 
the  line,  while  again  the  line  may  have  a  variable  angular  velocity 
about  the  point.     The  one  having  uniform  motion  of  the  point 
along  the  line,  and  uniform  angular  velocity  about  the  point 
is  called  the  Spiral  of  Archimedes. 

Fig.  152   shows   the   Archimedian   spiral   which   is   perhaps 
the    simplest    type    of    spiral.      If    ox    be 
assumed  as  the  primitive  position  of  a  line 
revolving  at  o,  and  the  point  also  starts  at 
o,  then  o  is  the  starting  point  of  the  curve.     x_ 
Suppose  that  after  one-eighth  of   a  revo- 
lution the  generating  point  has  moved  a 
distance  oa,  then,  after    one-quarter  of    a 
revolution,  the  point  will   be   at  b,  where 
ob  =  2Xoa,  and  so  on.     The  spiral  becomes  j?IG  152. 

larger  and  larger   as  the    revolution   con- 
tinues.    Had  the  line  revolved  in  the  other  direction,  the  curve 
would  have  been  the  reverse  of  the  one  shown. 

913.  Doubly  curved   line.     A  doubly   curved  line  is  one 
whose  direction  is  continually  changing  and  whose  points  do 
not  lie  in  one  plane.     A  piece  of  wire  may  be  twisted  so  as  to 
furnish  a  good  example  of  a  doubly  curved  line. 

914.  Representation    of   doubly   curved    lines.     As   no 

single  plane  can  contain  a  doubly  curved  line,  it  becomes  neces- 
sary to  use  two  planes  and  project,  orthographically,  the  line 
on  them.  Familiarity  is  had  with  the  representation  of  points 
in  space  on  the  principal  planes.  The  line  may  be  conceived  as 
being  made  up  of  an  infinite  number  of  points  and  each  point 
can  be  located  in  space  by  its  projections. 


152  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

Fig.  153  represents  a  curve  ABCD,  shown  by  its  horizontal 
projection  abed  and  its  vertical  projection  a'b'c'd'.  Any  point 

on  the  curve,  such  as  B,  is  found 
by  erecting  perpendiculars  from  b 
and  b'  and  extending  them  to  their 
intersection;  this  will  be  the  point 
sought.  The  principal  planes  must 
be  at  right  angles  to  each  other  if 
it  is  desired  to  locate  a  point  by 
FlG  153  erecting  these  perpendiculars;  other- 

wise, the   curve  must  be  imagined 

from  its  projections  when  the  planes  are  revolved  into  coinci- 
dence, as  is  customary  in  orthographic  projection. 

915.  Helix.  The  helix  is  a  doubly  curved  line  described 
by  a  point  having  motion  around  a  line  called  the  axis,  and  in 
addition,  a  motion  along  it.  Unless  it  is  noted  otherwise,  the 
helix  will  be  considered  as  having  a  uniform  circular  motion 
around  the  axis  and  also  a  uniform  motion  along  it.  The  curve 
finds  its  most  extensive  application  on  that  type  of  screw  known 
as  a  machine  screw.  This  is  then  a  uniform  cylindrical  helix. 
Wood  screws  furnish  examples  of  conical  helices.*  The  helix 
is  also  frequently  used  in  making  springs  of  a  type  known  as 
helical  springs.! 

Fig.  154  shows  the  construction  of  the  helix.  Assume  that 
the  drawing  is  made  in  the  third  angle  of  projection;  the  plan  is 
therefore  on  top.  The  ground  line  is  omitted  because  the  dis- 
tance of  the  points  from  the  principal  planes  is  not  required, 
but  only  their  relative  location  to  each  other.  To  return,  1', 
2',  ...  B'  is  the  plan  (horizontal  projection)  of  the  helix,  showing 
the  circular  motion  of  the  point  around  the  axis  ab.  In  the 
elevation  (vertical  projection),  the  starting  point  of  the  curve  is 
shown  at  1.  If  the  point  revolves  in  the  direction  of  the  arrow, 
it  will,  on  making  one-eighth  of  a  revolution,  be  at  2'  in  plan 

*  The  distinction  might  be  made  between  cylindrical  and  conical  helices 
by  considering  the  curve  as  being  drawn  on  the  surface  of  a  cylinder  or  a  cone 
as  the  case  may  be. 

t  Helical  springs  are  frequently  although  incorrectly  called  spiral  springs. 
The  spiral  has  been  previously  denned;  and  a  spring  of  that  shape  is  a  spiral 
spring. 


CLASSIFICATION   OF  LINES 


153 


and  at  2  in  elevation.  After  a  quarter  of  a  revolution,  the  point 
is  at  3'  in  plan,  and  at  3  in  elevation.  The  position  3  is  its  extreme 
movement  to  the  right,  for  at  4,  the  point  has  moved  to  the  left, 
although  continually  upward  as  shown  in  the  elevation.  On 
completion  of  one  revolution,  the  point  is  at  9,  ready  to  proceed 
with  an  identical  curve  beyond  it. 

The  distance  between  any  point  and  the  position  of  the  point 
after  one  complete  revolution  is  known  as  the  pitch.  The  dis- 
tance p  is  that  pitch,  and  may  be  given  from  any  point  on 
the  curve  to  the  succeeding  position  of  that  point  after  one 


FIG.  154. 


FIG.  155. 


revolution.      The  distance  01',   02',   etc.  is  the    radius  of    the 
helix. 

Fig.  155  is  a  double  helix  and  consists  simply  of  two  distinct 
helices,  generated  so  that  the  starting  point  of  one  helix  is  just 
one-half  of  a  revolution  ahead  or  behind  the  other  helix.  The 
portion  of  the  helix  that  is  in  front  of  the  axis  is  shown  in  full; 
that  behind  the  axis  is  shown  dotted  to  give  the  effect  of  a  helix 
drawn  on  a  cylinder.  The  pitch  again  is  measured  on  any  one 
curve  and  requires  a  complete  revolution.  It  must  not  be 
measured  from  one  curve  to  a  similar  position  on  the  next  curve 
as  the  two  helices  are  entirely  distinct.  Fig.  155  will  show  this 
correctly.  Double,  and  even  triple  and  quadruple  helices  are 
used  on  screws;  they  are  simply  interwoven  so  as  to  be  equal 
distances  apart. 


154 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


916.  Classification  of  lines. 


Straight  Lines:      Unidirectional. 


Lines 


Curved  Lines; 


Singly  Curved  Lines:  Di- 
rection continually  chang- 
ing, but  always  in  a  single 
plane. 


Doubly  Curved  Lines:  Di- 
rection continually  chang- 
ing, but  not  in  any  one 
plane. 


circle 

ellipse 

parabola 

hyperbola 

cycloid 

epicycloid 

hypocycloid 

spiral 

etc. 

helix 
etc. 


FIG.  156. 


917.  Tangent.     A   tangent   is   the   limiting   position    of    a 
secant  as  the  points  of  secancy  approach  and  ultimately  reach 
coincidence.     Suppose,  in  Fig.  156,  pq  is  a  secant  to  the  curve 

and  that  it  revolves  about  p  as  a 
centre.  At  some  time,  the  secant 
pq  will  assume  some  such  position  as 
pr;  the  point  q  has  then  moved  to  r 
and  if  it  continues,  it  will  pass 
through  p  and  reach  s.  The  secant 
then  intersects  on  the  opposite  side 
of  p.  If  the  rotation  be  such  that 
q  passes  through  r  and  ultimately 
coincides  with  p,  then  this  limiting 

position  of  the  secant,  shown  as  pt,  is  the  tangent  to  the  curve 
and  p  is  the  point  of  tangency.  It  will  be  observed  that  the 
tangent  is  fixed  by  the  point  p,  and  the  direction  of  the  limiting 
position  of  the  secant. 

The  condition  of  tangency  is  a  mutual  relation.  That  is, 
the  curve  is  tangent  to  the  line  or  the  line  is  tangent  to  the  curve. 
Also,  two  or  more  curves  may  be  tangent  to  each  other  because 
the  tangent  line  may  be  considered  (at  the  point  of  tangency) 
as  the  direction  of  the  curve  (see  Art.  920). 

918.  Construction  of  a  tangent.     If  a  tangent  is  to  be 
drawn  to  a  curve  from  an  outside  point,  the  drafting  room  method 
is  to  use  a  ruler  of  some  sort  and  place  it  a  slight  distance  away 
from  the  point  and  then  revolve  it  until  it  nearly  touches  the 


CLASSIFICATION  OF  LINES  155 

curve;  a  straight  line  then  drawn  through  the  point  and  touching 
the  curve  will  be  the  required  tangent. 

If  the  problem  is  to  draw  a  tangent  at  a  given  point  on  a 
curve,  however,  the  quick  method  would  be  to  estimate  the 
direction  so  that  the  tangent  appears  to  coincide  with  as  much 
of  the  curve  on  one  side  as  it  does  on  the  other. 

A  more  accurate  method  of  constructing  the  tangent  at  a 
given  point  on  a  curve  is  shown  in  Fig.  157.  Let  a  be  the  desired 
point  of  tangency.  Draw  through  a,  the  secants  ab,  ac,  ad, 
the  number  depending  upon  the  degree  of  accuracy,  but  always 
more  than  here  shown.  With  a  as  a  centre,  draw  any  indefinite 
arc  eh,  cutting  the  prolongations  of  the  secants.  Lay  off  the 
chord  ab  =  ei,  ac  =  fj,  and  ad=hk,  but  this  etter  chord  is  laid 


FIG.  157. 

off  to  the  left  of  the  arc,  because  its  secant  cuts  the  curve  to  the 
left  of  the  desired  point  of  tangency.  A  smooth  curve  may 
now  be  drawn  through  i,  j,  and  k,  and  where  this  curve  inter- 
sects the  indefinite  arc  at  g,  draw  ga,  the  required  tangent.  The 
proof  of  this  is  quite  simple.  If  ab,  ac  and  ad  be  considered  as 
displacements  of  the  points  b,  c  and  d  from  the  positions  that 
they  should  occupy  when  the  secant  becomes  a  tangent,  then, 
when  these  points  approach  a,  so  as  ultimately  to  coincide  with 
it,  the  displacement  is  zero.  The  secant  then  becomes  a  tangent. 
The  curve  ijk  is  a  curve  of  displacements  from  efh,  the  indefinite 
arc.  Hence,  the  desired  tangent  must  pass  through  g,  as  it 
lies  on  the  curve  efh  and  its  displacement  from  that  curve  is 
therefore  zero. 

919.  To  find  the  point  of  tangency.    On  the  other  hand, 
if  the  tangent  is  drawn,  and  it  is  desired  to  find  the  point  of 


156 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


v 

V 


Oik 


FIG.  158. 


tangency,  the  problem  becomes  slightly  different.  Suppose  TT, 
Fig.  158,  is  the  given  tangent,  and  atb  is  the  given  curve.  It  is 
desired  to  find  the  point  of  tangency  t.  Draw  ab,  cd  and  ef, 

any  chords  parallel  to  TT.  Lay  off 
hg,  through  a,  perpendicular  to  TT, 
and  br,  through  b,  perpendicular 
also  to  TT.  Make  gh  =  qr  =  ab.  In 
the  other  cases,  make  ij=op  =  cd, 
and  kl=mn  =  ef.  Now  draw  a 
— T  smooth  curve  through  hjlnpr. 
Where  this  curve  intersects  TT  at 
t,  the  desired  point  of  tangency  is 
found.  Again,  the  proof  is  quite 
simple.  The  lengths  of  the  vertical 
lines  above  and  below  TT  are  equal 
to  the  lengths  of  the  corresponding 
chords.  As  the  chords  approach 

the  tangent,  they  diminish  in  length  and  ultimately  become 
zero.  Where  the  curve  crosses  the  tangent,  the  chord  length  is 
zero,  and,  hence,  must  be  the  point  of  tangency. 

920.  Direction   of  a    curve.     A  curve  continually  changes 
its  direction,  but  at  any  given  point  its  direction  is  along  the 
tangent  to  the  curve  by  definition.     It  is  proved  in  Mechanics, 
that  if  forces  act  on  a  particle  so  as  to  give  it  a  curved  motion, 
the  particle  will  fly  off  along  a  tangent  when  the  impressed  forces 
cease  to  act.* 

921.  Angle  between  curves.     The  angle  between  two  inter- 
secting curves  is  the  same  as  the  angles,  made  by  the  tangents  at 
the  point  of  intersection,  because  the  tangents  determine  the 
direction  of  the  curve  at  that  point.     Hence,  to  draw  smooth 
curves,  it  is  necessary  that  the  tangent  at  the  end  of  one  curve 
should  coincide  with  the  tangent  at  the  beginning  of  the  next 
curve. 

922.  Intersection  of  lines.    Two  lines  intersect  when  they 
have  a  point  in  common.     When  the  term  intersection  is  used  in 
connection  with  a  pair  of  straight  lines,  it  necessarily  implies 
that  the  lines  make  an  angle  with  each  other  greater  than  zero; 


*  See  Newton's  laws  of  motion  in  a  text-book  on  Physics. 


CLASSIFICATION  OF  LINES  157 

since,  otherwise,  if  there  is  a  zero  angular  relation,  the  lines  become 
coincident  and  have  all  points  in  common. 

When  a  line  meets  a  curve,  the  angle  between  the  line  and 
the  curve  at  the  point  of  intersection  is  the  same  as  the  angle 
between  the  line  and  the  tangent  to  the  curve  at  that  point. 
When  this  angle  of  intersection  becomes  zero,  however,  by 
having  the  line  coincide  with  the  tangent,  then  it  is  the  special 
case  of  intersection  known  as  tangency. 

Similarly  when  two  curves  intersect  their  angle  of  intersection 
is  the  same  as  the  angle  between  their  tangents  at  the  point  of 
intersection.  Also,  when  this  angle  becomes  zero  then  the  curves 
are  tangent  to  each  other. 

Thus,  in  order  to  differentiate  between  the  two  types  of 
intersection,  angular  intersection  means  intersection  at  an  angle 
greater  than  zero;  and,  as  a  consequence,  the  intersection  at  a 
zero  angle  is  known  as  tangential  intersection.  When  inter- 
section is  unmodified,  then  angular  intersection  is  implied. 

923.  Order  of  contact  of  tangents.  A  tangent  has 
been  previously  defined  as  the  limiting  position  of  a  secant,  as  the 
points  of  secancy  approach  and  ultimately  coincide  with  each 
other.  This  contact,  for  simple  tangency,  is  of  the  first  order. 
Two  curves  may  also  be  tangent  to  each  other  so  as  to  have  con- 
tact of  the  first  order,  as  for  example,  two  circles,  internally  or 
externally  tangent. 

Suppose  two  curves  acb  and  abe,  Fig.  159,  have  first  order 
contact  at  a,  and  cut  each  other  at  the 
point  b.  If  the  curve  abe  be  made  to 
revolve  about  the  point  a  as  a  centre, 
so  as  to  maintain  simple  tangency  *  and 
also  to  have  the  point  b  approach  a,  then 
at  some  stage  of  the  revolution  the  point 
b  can  be  made  to  coincide  with  a. 
Under  this  condition  there  is  tangency  FIG.  159. 

of  a  higher   order   because  three   points 

were  made  to  ultimately  coincide  with  each  other;  and  it  is 
called  second  order  of  contact.  It  is  possible  to  have  third 
order  of  contact  with  four  coincident  points;  and  so  on.  The 

*  In  order  to  make  this  a  rigid  demonstration,  the  centre  of  curvature 
•of  the  two  curves,  at  the  point  a  must  not  be  the  same.  See  Art.  924. 


158  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

order  of  contact  is  always  one  less  than  the  number  of  points 
that  approach  coincidence. 

924.  Osculating  circle.     Centre  of  curvature.     Let  abc, 

Fig.  160,  be  a  curve  and  b,  a  point  through  which  a  circle  gdbe 
passes,  cutting  the  curve  abc  in  three  points  d,b  and  e.  If  the 
diameter  of  the  circle  is  properly  chosen,  it  may  be  revolved 
about  b  as  a  centre  so  that  the  points  d 
and  e  will  both  approach  and  ultimately 
coincide  with  b  at  the  same  instant. 
This  position  of  the  circle  is  shown  as  bf . 
Hence,  the  circle  bf  is  tangent  to  the 
curve  abc,  and  is  of  the  second  order 
of  contact.  This  circle  is  the  osculating 
circle.  As  this  osculating  circle  must 
more  nearly  approach  the  curvature  of 
the  curve  abc  than  any  other  circle,  its 

radius  at  the  point  b  is  the  radius  of  curvature.  At  every  other 
point  on  the  curve,  there  is  a  new  osculating  circle,  of  a  new 
centre,  and  of  a  new  radius.  Thus,  the  osculating  circle  is  the 
second  order  tangent  circle  at  the  point;  and  the  radius  of 
curvature  may  be  denned  as  the  radius  of  the  osculating  circle 
through  the  point,  the  centre  of  curvature  being  the  centre  of  the 
osculating  circle. 

925.  Osculating  plane.     If  a  tangent  be  drawn  to  any  doubly 
curved  line,  an  infinite  number  of  planes  may  be  passed  so  as  to 
contain  the  tangent.     If  some  one  position  of  the  plane  be  selected 
so  as  to  contain  the  tangent  and  a  piercing  point  of  the  doubly 
curved  line  on  it,  then  by  proper  revolution  of  the  plane  the 
piercing  point  can  be  made  to  approach  and  ultimately  coincide 
with  the  point  of  tangency;    in  this  position,  the  plane  is  an 
osculating  plane. 

To  put  the  matter  differently,  suppose  it  is  desired  to  find 
the  osculating  plane  at  some  point  on  a  doubly  curved  line.  In 
this  case,  draw  a  pair  of  secants  to  the  doubly  curved  line  which 
intersect  at  the  given  point.  A  plane  passed  through  these 
secants  will  cut  the  doubly  curved  line  in  three  points.  As  the 
secants  approach  tangency,  the  plane  will  approach  osculation, 
and  this  osculating  plane  is  identical  with  that  of  the  former 
discussion  for  the  same  point  on  the  curve.  If  the  curve  under 


CLASSIFICATION  OF  LINES  159 

consideration  be  a  plane  curve,  then  the  secants  will  lie  in  the 
plane  of  the  curve,  and,  hence,  the  osculating  plane  of  the  curve 
will  be  the  plane  of  the  curve. 

926.  Point  of  inflexion.     Inflexional  tangent.      Assume 
a  curve  dae,  Fig.  161,  and  through  some  one  point  a  draw  the 
secant  be.     If  the  secant  be  revolved  about  the  point  a  so  that 
the  points  of  intersection  b  and  c  approach  a,  at  some  stage  of 
the  revolution  they  will  coincide  with 

it  at  the  same  instant  if  the  point  a 

is  properly  chosen.     The  point  a  must 

be  such  that  three  points  on  the  curve 

ultimately  coincide  at  the  same  instant, 

Further  than  this,  the  radius  of  curva-  FIG.  161. 

ture  (centre  of  the  osculating  circle) 

must  make  an  abrupt  change  from  one  side  of  the  curve  to  the 

other  at  this  point.     The  point  of  inflexion  therefore  is  a  point 

at  which  the  radius  of  curvature  changes  from  one  side  of  the 

curve  to  the  other.     The  inflexional  tangent  is  the  tangent  at  the 

point  of    inflexion.      It  may  also  be  noted  that  the  inflexional 

tangent  has  a  second  order  of  contact  (three  coincident  points) 

and  therefore  is  the  osculating  line  to  the  curve  at  the  point  of 

inflexion. 

927.  Normal.     A  normal  to  a  curve  is  a  perpendicular  to 
the  tangent  at  the  point  of  tangency.     The  principal  normal 
lies  in  the  osculating   plane.      As  an  infinite  number  of  normals 
may  be  drawn  to  the  tangent  at  the  point  of  tangency,  the  nor- 
mal may  revolve  about  the  tangent  so  as  to  generate  a  plane 
which  will  be  perpendicular  to  the  tangent  and  thus  establishes 
a  normal  plane.     In  the  case  of  a  circle,  the  radius  at  the  point 
of  tangency  is  normal  to  the  tangent;   in  such  cases,  the  tangent 
is  easily  drawn,  as  it  must  be  perpendicular  to  the  radius  at  the 
point  of  contact. 

928.  Rectification.    When  a  curve  is  made  to  roll  on  a  straight 
line,  so  that  no  slip  occurs  between  the  curve  and  the  line,  the 
distance  measured  on  the  line  is  equal  to  the  corresponding  length 
of  the  curve.     This  process  of  finding  the  length  of  the  curve  is 
called  rectification.     Commercially  applied,  the  curve  is  measured 
by  taking  a  divider  and  stepping  off  very  small  distances;    the 


160 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


number  of  steps  multiplied  by  the  distance  between  points  will 
give  approximately  the  length  of  the  curye.  It  must  be  noticed 
that  the  divider  measures  the  chord  distance,  instead  of  the 
arc  distance  and  is  therefore  always  less  than  the  actual  length  of 
the  curve,  but  when  the  distance  is  taken  small  enough,  the 
accuracy  of  the  final  result  is  proportional  to  the  care  taken  in 
making  the  measurement. 

929.  Involute  and  evolute.  When  a  tangent  rolls  about  a 
fixed  curve,  any  point,  on  the  tangent  describes  a  second  curve 

which  is  the  involute  of  the  first 
curve.  Fig.  162  shows  this  in 
construction.  Let  aceg  be  the 
fixed  curve,  and  ab,  be  the  posi- 
tion of  a  taut  string  that  is  wound 
on  the  curve  aceg.  If  a  pencil 
point  be  attached  to  the  string 
and  unwound,  the  pencil  point 
will  describe  the  curve  bdfh  which 
is  the  involute  of  the  curve  aceg. 
The  process  is  the  same  as  though 
the  tangent  revolved  about  the 
curve  aecg  and  some  point  on  the 

tangent  acted  as  the  generating  point.  At  a,  the  radius  is  ab; 
at  c,  the  radius  is  cd,  which  is  equal  to  ab  plus  the  rectified  arc 
ac  (length  of  string).  It  may  be  observed  that  the  curve  aceg 
is  the  curve  of  centres  for  the  curve  bdfh. 

If  the  string  be  lengthened  so  that  ai  is  the  starting  position, 
it  will  describe  the  curve  ijkl  which  again  is  an  involute.  This 
second  involute  is  parallel  to  the  first,  because  the  distance  between 
the  two  curves  measured  along  the  rolling  tangent  (or  radius  of 
curvature)  is  constant  between  the  two  curves. 

The  primitive  curve  aceg  is  the  evolute.  The  tangent  rolls 
on  the  evolute  and  any  point  on  it  describes  an  involute.  As 
any  point  on  the  tangent  will  answer  as  the  tracing  point,  it 
follows  that  every  evolute  has  an  infinite  number  of  involutes, 
all  of  which  are  parallel  curves. 

Reversing  the  process  of  the  construction  of  the  involute,  the 
method  of  drawing  the  primitive  curve  or  evolute  is  obtained. 
If  normals  are  drawn  to  the  involute,  consecutive  positions  of 


FIG.  162. 


CLASSIFICATION  OF  LINES  161 

the  normals  will  intersect.  The  locus  of  these  successive  inter- 
sections will  regenerate  the  primitive  curve  from  which  they 
have  been  evolved.* 

Again,  the  length  of  the  tangent  to  the  e volute  is  the  radius 
of  curvature  for  the  involute.  As  the  radius  of  any  circle  is 
perpendicular  to  the  tangent  at  that  point,  it  follows  that  the 
involute  is  always  normal,  point  for  point,  to  the  evolute,  since 
the  rolling  tangent  is  the  direction  of  the  evolute  at  the  point 
of  contact  and  that  again  is  normal  to  the  involute. 

930.  Involute  of  the  circle.  The  involute  of  the  circle 
is  a  plane  curve,  described  by  a  point  on  a  tangent,  while  the 
tangent  revolves  about  the  circle. 

Let  o,  Fig.  163,  be  the  centre  of  a  circle  whose  radius  is  oa. 
Let,  also,  a  be  the  starting  point  of 
the  involute.  Divide  the  circle  in 
any  number  of  parts,  always,  however, 
more  than  are  shown  in  the  illustra- 
tion.  Draw  tangents  to  the  various 
radii.  On  them,  lay  off  the  rectified 
arc  of  the  circle  between  the  point  of 
tangency  and  the  starting  point.  For  FIG  163 

instance,  eb  equals  the  rectified  length 

of  the  arc  ea;  also  fc  equals  the  rectified  semicircle  fea;  gd 
equals  the  rectified  length  of  the  arc  gfea.  The  involute  may 
be  continued  indefinitely  for  an  infinite  number  of  revolutions, 
but,  discussion  is  usually  centred  on  some  limited  portion. 

The  curve  here  shown  is  approximately  the  same  as  that 
described  by  the  end  of  the  thread,  when  a  spool  is  unwound. 

QUESTIONS  ON  CHAPTER  IX 

1.  Why  are  lines  and  points  considered  as  mathematical  concepts? 

2.  How  is  a  straight  line  denned? 

3.  By  what  two  means  may  a  straight  line  be  fixed  in  space? 

4.  What  is  a  generating  point? 

5.  What  is  a  locus? 

6.  What  is  a  singly  curved  line? 

7.  What  is  a  plane  curve? 

8.  How  are  plane  curves  represented? 

*  The  evolute  of  a  circle,  therefore,  is  its  centre. 


162  GEOMETRICAL  PROBLEMS  IN   PROJECTION 

9.  Show  the  mode  of  representing  curves  when  their  planes  are  parallel 
to  the  plane  of  projection?    When  perpendicular?     When  inclined? 

10.  Is  it  desirable  to  use  the  plane  of  the  curve  as  the  plane  of  projection? 

11.  Define  the  circle. 

12.  What  is  the  radius?     Diameter?    Sector?    Segment? 

13.  When  the  chord  passes  through  the  centre  of  the  circle,  what  does 

it  become? 

14.  What  is  a  secant?     Quadrant?     Semicircle? 

15.  Define  concentric  circles;  eccentric  circles;  eccentricity. 

16.  What  is  an  ellipse? 

17.  Define  major  axis  of  an  ellipse;  n:inor  axis;  foci. 

18.  Describe  the  accurate  method  of  c  rawing  an  ellipse  by  the  intersection 

of  circular  arcs. 

19.  Describe  the  trammel  method  of  drawing  an  ellipse. 

20.  Draw  an  ellipse  whose  major  axis  is  3"  long  and  whose  minor  axis 

is   2"   long.     Use   the   accurate   method   of   intersecting   circular 
arcs. 

21.  Construct  an  ellipse  whose  major  axis  is  3"  long  and  whose  minor 

axis  is  1|"  long.     Use  the  trammel  method. 

22.  What  is  a  parabola? 

23.  Define  focus  of  a  parabola;  directrix;  axis;  vertex. 

24.  Is  the  parabola  symmetrical  about  the  axis? 

25.  Is  the  parabola  an  open  or  a  closed  curve? 

26.  Construct  the  parabola  whose  focus  is  2"  from  the  directrix. 

27.  What  is  a  hyperbola? 

28.  Define  foci  of  hyperbola;  transverse  axis;  conjugate  axis;  vertex; 

asymptote. 

29.  How  many  branches  has  a  hyperbola?    Are  they  symmetrical  about 

the  transverse  and  conjugate  axes? 

30.  How  many  asymptotes  may  be  drawn  to  a  hyperbola? 

31.  Is  the  hyperbola  an  open  or  a  closed  curve? 

32.  Construct  a  hyperbola  whose  distance  between  foci  is  2"  and  whose 

constant  difference  is  \" . 

33.  What  is  a  cycloid? 

34.  Define  rolling  circle  of  a  cycloid;  directrix. 

35.  Construct  a  cycloid  whose  diameter  of  rolling  circle  is  \\" .     Draw 

the  curve  for  one  revolution  only. 

36.  What  is  an  epicycloid? 

37.  What  form  of  directrix  has  the  epycycloid? 

38.  Construct  the  epicycloid,  whose  diameter  of  rolling  circle  is  \\" 

and  whose  diameter  of  directrix  is  8".     Draw  the  curve  for  one 
revolution  only. 

39.  What  is  a  hypocycloid? 

40.  What  is  the  form  of  the  directrix  of  the  hypocycloid? 

41.  Construct  a  hypocycloid  whose  diameter  of  rolling  circle  is  2"  and 

whose  diameter  of  directrix  is  9".    Draw  the  curve  for  one  revolu- 
tion only. 

42.  What  is  a  spiral? 


CLASSIFICATION  OF  LINES  163 

43.  Construct  an  Archimedian  spiral  which  expands  1|"  in  a  complete 

revolution.     Draw  the  spiral  for  two  revolutions. 

44.  What  is  a  doubly  curved  line? 

45.  Draw  a  doubly  curved  line  with  the  principal  planes  in  oblique 

projection. 

46.  Construct  the  orthographic  projection  from  Question  45. 

47.  What  is  a  helix? 

48.  Define  uniform  cylindrical  helix;  conical  helix;  diameter  of  helix; 

pitch. 

49.  Construct  a  helix  whose  diameter  is  2",  and  whose  pitch  is  I". 

Draw  for  two  revolutions. 

50.  Construct  a  triple  helix  whose  diameter  is  2"  and  whose  pitch  is  3". 

The  helices  are  spaced  equally  and  are  to  be  drawn  for  1|  revolu- 
tions. 

51.  Make  a  classification  of  lines. 

52.  What  is  a  tangent? 

53.  Is  the  tangent  fixed  in  space  by  a  point  and  a  direction? 

54.  Show  that  the  tangent  is  the  limiting  position  of  a  secant. 

55.  How  is  a  tangent  drawn  to  a  curve  from  a  point  outside? 

56.  Given  a  curve  and  a  point  on  it,  draw  a  tangent  by  the  accurate 

method.     Prove  that  the  chord  length  is  zero  for  the  tangent 
position. 

57.  Given  a  curve  and  a  tangent,  determine  the  point  of  tangency.    Prove. 

58.  What  is  the  direction  of  a  curve? 

59.  What  is  the  angle  between  two  intersecting  curves? 

60.  When  several  curves  are  to  be  joined,  show  what  must  be  done  to 

make  them  smooth  curves. 

61.  Define  intersection  of  lines. 

62.  Show  that  the  tangent  intersects  the  curve  at  a  zero  angle. 

63.  Define  order  of  contact  of  tangents. 

64.  If  three  points  become  coincident  on  tangency,  what  order  of  contact 

does  the  tangent  have? 

65.  Define  osculating  circle. 

66.  Define  centre  of  curvature. 

67.  Is  the  radius  of  the  osculating  circle  to  a  curve  the  radius  of  curvature 

at  that  point? 

68.  Show  how  an  osculating  circle  may  have  second  order  contact  with 

a  plane  curve. 

69.  What  is  an  osculating  plane? 

70.  When  is  the  osculating  plane  the  plane  of  the  curve? 

71.  Define  point  of  inflexion. 

72.  What  is  an  inflexional  tangent? 

73.  Does  the  radius  of  curvature  change  from  one  side  of  the  curve  to 

the  other  at  a  point  of  inflexion? 

74.  Define  principal  normal. 

75.  Show  that  the  centres  of  curvature  at  a  point  of  inflexion  lie  on 

opposite  sides  of  the  normal  to  the  curve  through  the  point  of 
inflexion. 


164  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

76.  When  a  normal  is  drawn  to  a  curve  is  the  one  in  the  osculating  plane 

the  one  generally  understood? 

77.  How  many  normals  may  be  drawn  to  a  doubly  curved  line  at  a 

given  point? 

78.  What  is  meant  by  rectification? 

79.  Rectify  a  2"  diameter  circle.     Compute  the  length  of  the  rectified 

circle  (=2x3.1416)  and  express  the  ratio  of  rectified  to  computed 
length  as  a  percentage. 

80.  Define  involute  and  evolute. 

81.  Show  that  all  involutes  to  a  curve  are  parallel  curves. 

82.  Show  that  the  involute  is  always  normal  to  the  evolute  at  the  point 

for  which  it  corresponds. 

83.  Show  that  the  drawing  of  the  evolute  is  the  reverse  process  of  drawing 

the  involute. 

84.  Draw  the  involute  of  a  circle. 

85.  Draw  the  involute  to  an  ellipse. 

86.  Draw  the  evolute  to  an  ellipse. 

87.  Draw  the  involute  to  a  parabola. 

88.  Draw  the  evolute  to  a  parabola. 

89.  Draw  the  involute  to  one  branch  of  a  hyperbola. 

90.  Draw  the  evolute  to  one  branch  of  a  hyperbola. 

91.  Draw  the  involute  to  a  cycloid. 

92.  Draw  the  evolute  to  a  cycloid. 

93.  Draw  the  involute  to  an  epicycloid. 

94.  Draw  the  evolute  to  an  epicycloid. 

95.  Draw  the  involute  to  a  hypocycloid. 

96.  Draw  the  evolute  to  a  hypocycloid. 

97.  Draw  the  involute  to  an  Archimedian  spiral. 

98.  Draw  the  evolute  to  an  Archimedian  spiral. 


CHAPTER  X 

CLASSIFICATION  OF  SURFACES 

1001.  Introductory.     A  surface  may  be  generated  by  the 
successive  positions  of  a  line  which  moves  so  as  to  generate  an 
area.     As  there  are  infinite  varieties  of  lines  and  as  their  motion 
may  again  be  in  an  infinite  variety  of  ways,  therefore,  an  infinite 
variety  of  possible  surfaces  result.     In  engineering  it  is  usual 
to  limit  the  choice  of  surfaces  to  such  as  may  be  easily  reproduced 
and    easily    represented.     Surfaces,   like    lines    and    points,    are 
mathematical   concepts   because   they   have   no    material   exist- 
ence. 

When  curved  surfaces,  of  a  more  or  less  complex  nature,  are 
to  be  represented,  they  may  be  shown  to  advantage,  by  the 
effects  of  light  on  them.  Examples  of  this  kind  are  treated  in 
Chapters  XIV  and  XV. 

1002.  Plane  Surface.     A  plane   surface   is   a   surface   such 
that  when  any  two  points  in  it  are  joined  by  a  straight  line,  the 
line  lies  wholly  within  the  surface.     Thus,  three  points  may  be 
selected  in  a  plane  and  two  intersecting  lines  may  be  drawn 
through  the  three  points;  the  intersecting  lines  lie  in  the  plane 
and,  therefore,  may  be  used 'to  determine  it.     Also,  a  line  and 
an  external  point  may  determine  a  plane. 

The  plane  surface  may  also  be  conceived  as  being  generated 
by  a  straight  line,  moving  so  as  to  touch  another  line,  and  con- 
tinually remaining  parallel  to  its  original  position.  Hence, 
also,  two  parallel  lines  determine  a  plane. 

In  the  latter  case,  the  moving  straight  line  may  be  consid- 
ered as  a  rectilinear  generatrix,  touching  a  rectilinear  directrix, 
and  occupying  consecutive  positions  in  its  motion.  Any  one 
position  of  the  generatrix  may  be  used  as  an  element  of  the 
surface. 

165 


166 


GEOMETRICAL  PROBLEMS  IX  PROJECTION 


Fig.  164  shows  a  plane  surface  ABCD  on  which  straight  lines 
ab,  cd,  ef  and  gh  are  drawn,  all  of  which  must  lie  wholly  within 
the  plane,  irrespective  of  the  direction  in  which  they  are  drawn. 
Any  curve  drawn  on  this  surface  is  a  plane  curve. 

1003.  Conical  surface.  If  a  straight  line  passes  through 
a  given  point  in  space  and  moves  so  as  to  touch  a  given  fixed 
curve,  the  surface  so  generated  is  a  conical  surface.  The  straight 
line  is  the  rectilinear  generatrix,  the  fixed  point  is  the  vertex 
and  the  given  fixed  curve  is  the  directrix,  which  need  not  be  a 
closed  curve.  The  generatrix  in  any  one  position  is  an  element 
of  the  surface. 

Fig.  165  shows  a  conical  surface,  generated  in  the  manner 


FIG.  164. 


FIG.  165. 


indicated.  Either  the  upper  or  the  lower  curve  may  be  consid- 
ered as  the  directrix.  In  fact,  any  number  of  lines  may  be  drawn 
on  the  resulting  surface,  whether  the  lines  be  singly  or  doubly 
curved,  and  any  of  which  will  fill  the  office  of  directrix.  A  plane 
curve  is  generally  used  as  the  directrix. 

With  a  generatrix  line  of  indefinite  extent,  the  conical  sur- 
face generated  is  a  single  surface  (not  too  surfaces  as  might 
appear) ;  the.  vertex  o  is  a  point  of  union  and  not  of  separation. 
The  portion  of  the  surface  from  the  vertex  to  either  side  is  called 
a  nappe;*  hence,  there  are  two  nappes  to  a  conical  surface. 

1004.  Cone.    The  cone  is  a  solid,  bounded  by  a  closed  conical 
surface  of  one  nappe  and  a  plane  cutting  all  the  elements.    The 
*  Pronounced  "nap." 


CLASSIFICATION  OF  SURFACES 


167 


curve  of  intersection  of  the  rectilinear  elements  and  the  plane 
cutting  all  the  elements  is  the  base.  A  circular  cone  has  a  circle 
for  its  base  and  the  line  joining  the  vertex  with  the  centre  of  the 
base  is  the  axis  of  the  cone.  If  the  axis  is  perpendicular  to  the 
plane  of  the  base  the  cone  is  a  right  cone.  When  the  base  is  a 
circle,  and  the  axis  is  perpendicular  to  the  plane  of  the  base, 
the  cone  is  a  right  circular  cone  or  a  cone  of  revolution.*  A 
cone  of  revolution  may  be  generated  by  revolving  a  right  tri- 
angle about  one  of  its  legs  as  an  axis.  The  hypothenuse  is  then 
the  slant  height  of  the  cone.  The  perpendicular  distance  from 
the  vertex  to  the  plane  of  the  base  is  the  altitude  of  the  cone. 
The  foot  of  the  perpendicular  may  fall  outside  of  the  centre  of 
the  base  and  in  such  a  case,  the  cone  is  an  oblique  cone. 

The  frustum  (plural : — frusta)  of  a  cone  is  the  limited  portion 
of  the  solid  bounded  by  a  closed  conical  surface  and  two  parallel 
planes,  each  cutting  all  the  elements,  and  giving  rise  to  the  upper 
base  and  the  lower  base  of  the  frustum  of  a  cone.  The  terms 
upper  and  lower  base  are  relative;  it  is  usual  to  consider  the 
larger  as  the  lower  base  and  to  represent  the  figure  as  resting 
on  it.  When  the  cutting  planes  are  not  parallel,  then  the  solid 
is  a  truncated  cone. 

1005.  Representation  of  the  cone.  A  cone,  like  any 
other  object,  is  represented  by  its  projec- 
tions on  the  principal  planes.  For  con- 
venience in  illustrating  a  cone,  the  plane 
of  the  base  is  assumed  perpendicular  to 
one  of  the  principal  planes,  as  then  its 
projection  on  that  plane  is  a  line.  Fig. 
166  shows  a  cone  in  orthographic  projec- 
tion. The  vertex  O  is  shown  by  its 
projections  o  and  o';  dV  is  the  vertical 
projection  of  the  base  since  the  plane  of 
the  base  is  assumed  perpendicular  to  the 
vertical  plane.  The  extreme  limiting  ele- 
ments oV  and  o'd'  are  also  shown,  thus 

completing  the  vertical  projection.     In  the  horizontal  projection, 
any  curve   acbd  is   assumed  as   the   projection  of   the  base  so 

*  This  distinction  is  made  because  a  cone  with  an  elliptical  base  may 
also  be  a  right  cone  when  the  vertex  is  chosen  so  that  it  is  on  a  perpendicular 
to  the  plane  of  the  base,  at  the  intersection  of  the  major  and  minor  axes. 


FIG.  166. 


168 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


that  d  and  c  are  corresponding  projections  of  d'  and  c'.  From 
o,  the  lines  ob  and  oa  are  drawn,  tangent  to  acbd,  thus  completing 
the  horizontal  projection. 

It  must  here  be  emphasized,  that  acbd  is  not  the  actual  base 
of  the  cone,  but  only  its  projection.  It  is  impossible  to  assume 
two  curves,  one  in  each  plane  of  projection,  and  call  them  corre- 
sponding projections  of  the  same  base.  The  corresponding  points 
must  be  selected,  so  that  they  will  lie  in  one  plane,  and  that 
plane  must  be  the  plane  of  the  base.  If  it  be  desired  to  show 
the  base  in  both  projections,  when  the  plane  of  the  base  is  in- 
clined to  the  principal  planes,  it  is  necessary  to  assume  one  pro- 
jection of  the  base.  Lines  are  then  drawn  in  that  plane,  through 
the  projection  of  the  base  and  the  corresponding  'projections 


FIG.  167. 


FIG.  168. 


of  the  lines  are  found.  The  points  can  then  be  determined  as 
they  must  be  situated  on  these  lines.  (Arts.  704  and  811.) 

1006.  To  assume  an  element  on  the  surface  of  a  cone. 

To  assume  an  element  of  a  cone,  assume  the  horizontal  projec- 
tion oa  in  Fig.  167.  There  are  two  elements  on  the  cone  which 
have  the  same  horizontal  projection  and  they  are  shown  as 
o'a'  and  o'b'  in  the  vertical  projection.  If  oa  be  considered 
visible,  while  viewing  the  horizontal  plane,  then  o'a'  is  its  corre- 
sponding projection.  If  o'b'  be  the  one  assumed  projection 
then  oa  is  on  the  far  side  and  should  in  this  case  be  drawn  dotted. 

1007.  To  assume  a  point  on  the  surface  of  a  cone.     To 

assume  a  point  on  the  surface  of  a  cone,  assume  c'  in  Fig.  168 
as  the  vertical  projection,  somewhere  within  the  projected  area. 


CLASSIFICATION  OF  SURFACES 


169 


Draw  the  element  o'b'  through  c'  and  find  the  corresponding, 
projection  of  the  element.  If  o'b'  is  visible  to  the  observer, 
then  ob  is  the  corresponding  projection,  and  c  on  it  is  the  required 
projection.  If  o'b'  is  on  the  far  side,  then  d  is  the  desired  pro- 
jection. 

A  slightly  different  case  is  shown  in  Fig.  169.  If  c  is  assumed 
on  the  visible  element  oa  then  c'  is  the  corresponding  projection 
on  o'a'.  Otherwise,  if  ob  is  dotted  (invisible)  then  o'b'  is  the 
corresponding  element  and  d  and  d'  are  corresponding  projections. 

1008.  Cylindrical  surface.  When  a  straight  line  moves 
so  that  it  remains  continually  parallel  to  itself  and  touches  a 
given  fixed  curve,  the  surface  generated  is  a  cylindrical  surface. 


FIG.  169. 


FIG.  170. 


The  straight  line  is  the  rectilinear  generatrix,  the  fixed  curve 
is  the  directrix  and  need  not  be  a  closed  curve.  The  generatrix 
in  any  one  position  is  an  element  of  the  surface. 

Fig.  170  shows  a  cylindrical  surface,  generated  in  the  manner 
indicated.  Any  curve,  drawn  on  the  resultant  surface,  whether 
singly  or  doubly  curved,  may  be  considered  as  the  directrix. 
The  limiting  curves  that  are  shown  in  the  figure  may  also  be  used 
as  directrices.  A  plane  curve  is  generally  used  as  a  directrix 

1009.  Cylinder.  A  cylinder  is  a  solid  bounded  by  a  closed 
cylindrical  surface  and  two  parallel  planes  cutting  all  the  elements. 
The  planes  cut  curves  from  cylindrical  surface  which  form  the 
bases  of  the  cylinder  and  may  be  termed  upper  and  lower  bases 
if  the  cylinder  is  so  situated  that  the  nomenclature  fits.  When 
the  planes  of  the  bases  are  not  parallel  then  it  is  called  a  truncated 
cylinder. 


170  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

Should  the  bases  have  a  centre,  a  figure  such  as  a  circle  for 
instance,  then  a  line  joining  these  centres  is  the  axis  of  the  cylinder. 
The  axis  must  be  parallel  to  the  elements  of  the  cylinder.  If  the 
axis  is  inclined  to  the  base,  the  cylinder  is  an  oblique  cylinder. 
On  the  other  hand,  of  the  axis  is  perpendicular  to  the  plane  of 
the  base,  it  is  a  right  cylinder  and  when  the  base  is  a  circle,  it  is 
a  right  circular  cylinder,  or,  a  cylinder  of  revolution.  The 
cylinder  of  revolution  may  be  generated  by  revolving  a  rectangle 
about  one  of  its  sides  as  an  axis.  A  right  cylinder  need  not  have 
a  circular  base,  but  the  elements  must  be  perpendicular  to  the 
plane  of  the  base. 

1010.  Representation     of     the     cylinder.      A     cylinder, 
represented   orthographically  is   shown   in   Fig.    171.      Suppose 
the  base  is  assumed  in  the  horizontal  plane,  then  e'g'    may    be 

taken  as  the  vertical  projection  of  the  base. 
Also  e'f  and  g'h',  parallel  to  each  other, 
may  be  taken  as  the  projections  of  the 
extreme  limiting  elements.  Any  curve,  as 
aecg,  may  be  drawn  for  the  horizontal 
projection  so  long  as  e  and  g  are  corre- 
sponding projections  of  e'  and  g'.  In  addi- 
tion, draw  ab  and  cd  parallel  to  each  other 
and  tangent  to  the  curve  aecg.  The  hori- 
zontal projection  is  thus  completed.  It  is 
to  be  noted  that  aecg  is  the  true  base 
FIG.  171.  because  it  lies  in  the  horizontal  plane.  If 

the    plane  of    the  base  does    not  coincide 

with  the  horizontal  plane,  then,  as  in  Art.  1005,  what  applies 
to  the  selection  of  the  projection  of  the  base  of  the  cone  applies 
here. 

1011.  To  assume  an  element  on  the  surface  of  a  cylinder. 

To  assume  an  element  on  the  surface  of  a  cylinder,  select  any 
line,  ab,  Fig.  172,  as  the  horizontal  projection.  As  all  parallel 
lines  have  parallel  projections,  then  ab  must  be  parallel  to  the 
extreme  elements  of  the  cylinder.  If  ab  is  assumed  as  a  visible 
element,  then  a'b'  is  its  corresponding  projection,  and  is  shown 
dotted,  because  hidden  from  view  on  the  vertical  projection. 
If  c'd'  is  a  visible  element,  then  cd  should  be  dotted  in  the 
horizontal  projection. 


CLASSIFICATION  OF  SURFACES 


171 


1012.  To  assume  a  point  on  the  surface  of  a  cylinder. 

To  assume  a  point  on  the  surface  of  a  cylinder,  select  any  point 
c,  Fig.  173,  in  the  horizontal  projection,  and  draw  the  element 
ab  through  it.  Find  the  corresponding  projection  a'b'  and  on 
it,  locate  c',  the  required  projection.  What  has  been  said  before 
(Art.  1005)  about  the  two  possible  cases  of  an  assumed  projection, 
applies  equally  well  here  and  should  require  no  further  mention. 

1013.  Convolute  surface.     A  convolute  surface  is  a  surface 
generated  by  a  line  which  moves  so  as  to  be  continually  tangent  * 
to   a   line   of   double   curvature.     For   purposes   of   illustration, 
the  uniform  cylindrical  helix  will  be  assumed  as  the  line  of  double 


11    13 


FIG.  172. 


FIG.  173. 


FIG.  174. 


curvature,  remembering,  in  all  cases,  that  the  helix  may  be 
variable  in  radius  and  in  motion  along  the  axis,  so  that  its  char- 
acteristics may  be  imparted  to  the  resulting  convolute. 

The  manner  in  which  this  surface  is  generated  may  be  gained 
from  what  follows.  Let  abed,  Fig.  174,  be  the  horizontal  pro- 
jection of  a  half  portion  octagonal  prism  on  which  is  a  piece  of 
paper  in  the  form  of  a  right  triangle  is  wound.  The  base  of  the 
triangle  is  therefore  the  perimeter  of  the  prism  and  the  hypoth- 
enuse  will  appear  as  a  broken  line  on  the  sides  of  the  prism. 
If  the  triangle  be  unwound  from  the  prism  and  the  starting  point 
of  the  curve  described  by  the  hypothenuse  on  the  horizontal 

*  It  may  be  observed  that  the  tangent  to  a  line  of  double  curvature  must 
ile  in  the  osculating  plane  (Art.  925). 


172 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


plane  be  at  o,  then  the  portion  of  the  triangle  whose  base  is  oa 
will  revolve  about  the  edge  a  so  as  to  describe  the  arc  ol.  At 
the  point  1  the  triangle  is  free  along  the  face  ab  and  now  swings 
about  b  as  a  centre  and  describes  the  arc 
1-2.  As  the  process  goes  on  to  the  point 
2,  the  triangle  is  free  on  the  face  be  and 
then  swings  about  c  as  a  centre  and 
describes  the  arc  2-3;  and  so  on.  In  the 
vertical  projection,  the  successive  positions 
of  the  hypothenuse  are  shown  by  a'l', 
b'2',  c'3'  etc.  It  will  be  noted  that  a'l' 
intersects  b'2'  at  b',  and  b'2'  intersects 
c'3'  at  c',  etc.;  but,  a'l'  does  not  intersect 
c'3'  nor  does  b'2'  intersect  d'4'.  Hence,  the 
elements  of  the  surface  generated  by  the 
hypothenuse  intersect  two  and  two.  The 
first  element  intersects  the  second;  the 
second  the  third;  the  third  the  fourth,  etc.; 

but  the  first  does  not  intersect  the  third,  or  any  beyond,  nor  does 
the  second  intersect  the  fourth  or  any  elements  beyond  the  fourth. 
When  the  prism  approaches  a  cylinder  as  a  limit  by  increas- 
ing the  number   of  sides  indefinitely,  the   hypothenuse  wound 


FIG.  174. 


FIG.  175. 


FIG.  176. 


around  the  cylinder  approaches  a  helix  as  a  limit;  the  unwind- 
ing hypothenuse  will  become  the  generatrix,  tangent  to  the  helix, 
and  will  approach  the  desired  convolute  surface.  The  ultimate 
operation  is  a  continuous  one  and  may  be  seen  in  Fig.  175.  The 
curve  abed  described  by  the  hypothenuse  fd  on  the  plane  MM 


CLASSIFICATION  OF  SURFACES 


173 


FIG.  177. 


is  the  involute  of  a  circle,  if  the  cylinder  is    a  right  circular 
cylinder. 

Fig.  176  may  indicate  the  nature  of  the  surface  more  clearly. 
Examples  of  this  surface  may  be 
obtained  in  the  machine  shop  on 
observing  the  spring-like  chips, 
that  issue  on  taking  a  heavy  cut 
from  steel  or  brass.  The  surfaces 
are  perhaps  not  exact  convolutes, 
but  they  resemble  them  enough  to 
give  the  idea. 

It  is  not  necessary  to  have  the 
tangent  stop  abruptly  at  the 
helix,  the  tangent  may  be  a  line 
of  indefinite  extent,  and,  hence, 
the  convolute  surface  extends  both 
sides  of  the  helical  directrix.  No 
portion  of  this  surface  intersects 
any  other  portion  of  the  surface, 
but  all  the  convolutions  are  dis- 
tinct from  each  other.  Fig.  177  will  perhaps  convey  the  final 
idea. 

1014.  Oblique  helicoidal  screw  surface.*  When  the  helical 
directrix  of  a  convolute  surface  decreases  in  diameter,  it  will 

ultimately  coincide  with  the  axis 
and  the  helix  will  become  a  line. 
The  oblique  helicoidal  screw  sur- 
face, therefore,  resolves  itself  into 
FIG.  178.  tne  surface  generated  by  a  recti- 

linear   generatrix   revolving    about 

another  line  which  it  intersects,  at  a  constant  angle,  the  inter- 
section moving  along  the  axis  at  a  uniform  rate.  The  application 
of  this  surface  is  shown  in  the  construction  of  the  V  thread 
screw  which  in  order  to  become  the  United  States  Standard 
screw,  must  make  an  angle  of  60°  at  the  V  as  shown  in  Fig.  178. 

*  The  helicoid  proper  is  a  warped  surface  (1016).  If  a  straight  line 
touches  two  concentric  heb'ces  of  different  diameters  and  lies  in  a  plane  tangent 
to  the  inner  helix's  cylinder,  the  line  will  generate  a  warped  surface.  When 
the  diameter  of  the  cylinder  becomes  zero,  the  helix  becomes  a  line  and  the 
helicoidal  surface  is  the  same  as  that  here  given. 


174 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


FIG.  179. 


1015.  Right  helicoidal    screw  surface.      If  the   diameter 
of  the  helical  directrix  still  remains  zero,   and  the  rectilinear 

generatrix  becomes  perpendicular  to 
the  axis  and  revolves  so  that  the  in- 
tersection of  the  axis  and  the  genera- 
trix moves  along  the  axis  at  a  constant 
rate,*  a  special  case  of  the  convolute  is 
obtained.  This  special  case  of  the 
convolute  is  called  a  right  helicoidal 
screw  surface,  and  when  applied  gives 
the  surface  of  a  square  threaded  screw  as  shown  in  Fig.  179. 

In  both  cases  of  the  oblique  and  right  helicoidal  screw  sur- 
faces the  helices  at  the  outside  and  root  (bottom)  of  the  thread, 
are  formed  by  the  intersection  of  the  screw  surface  and  the 
outer  and  inner  concentric  cylinders.  The  pitch  of  both  must 
be  the  same,  as  every  point  on  the  generatrix  advances  at  a 
uniform  rate.  Hence,  the  angle  of  the  tangent  to  the  helix 
must  vary  on  the  inner  and  outer  cylinders.  For  this  reason, 
the  helices  have  a  different  shape  notwithstanding  their  equal 
pitch. 

1016.  Warped  surface.  A  warped  surface  is  a  curved 
surface,  generated  by  a  rectillinear  generatrix,  moving  so  that 
no  two  successive  elements  lie  in  the  same  plane.  Thus,  the 
consecutive  elements  can  neither  be  parallel  nor  intersect,  hence, 
they  are  skew  lines.  An  example  of  this  surface  may  be  obtained 
by  taking  a  series  of  straight  sticks  and  drilling  a  small  hole  through 
each  end  of  every  stick.  If  a  string  be  passed  through  each  end 
and  secured  so  as  to  keep  them  together,  the  series  of  sticks 
may  be  laid  on  a  flat  surface  and  thus  represent  successive 
elements  of  a  plane.  It  may  also  be  curved  so  as  to  represent 
a  cylinder.  Lastly,  it  may  be  given  a  twist  so  that  no  single 
plane  can  be  passed  through  the  axis  of  successive  sticks;  this 
latter  case  would  then  represent  a  warped  surface. 

Warped  surfaces  find  comparatively  little  application  in 
engineering  because  they  are  difficult  to  construct  or  to  duplicate. 
At  times,  however,  they  are  met  with  in  the  construction  of 

*  If  the  pitch  becomes  zero  when  the  diameter  of  the  helix  becomes  zero, 
it  is  the  case  of  a  line  revolving  about  another  line,  through  a  fixed  point; 
the  surface  is  therefore  a  cone  of  revolution,  if  the  generatrix  is  inclined 
to  the  axis.  If  the  generatrix  is  normal  to  the  axis,  the  surface  is  a  plane. 


CLASSIFICATION  OF  SURFACES  175 

"  forms  "  for  reinforced  concrete  work,  where  changes  of  shape 
occur  as  in  tunnels,  and  similar  constructions;  in  propeller  screws 
for  ships;  in  locomotive  "  cow-catchers/'  etc. 

1017.  Tangent  plane.     If  any  plane  be  passed  through  the 
vertex  of  a  cone,  it  may  cut  the  surface  in  two  rectilinear  elements 
under  which  condition  it  is  a  secant  plane.     If  this  secant  plane 
be  revolved  about  one  of  the  rectilinear  elements  as  an  axis, 
the  elements  of  secancy  can  be  made  to  approach  so  as  to  coincide 
ultimately.     This,  then,  is  a  limiting  position  of  the  secant  plane, 
in  which  case  it  becomes  a  tangent  plane,  having  contact  with 
the  cone  all  along  one  element. 

If  through  some  point  on  the  element  of  contact,  two  inter- 
secting curved  lines  be  drawn  on  the  surface  of  the  cone,  then, 
also,  two  secants  may  be  drawn  to  these  curved  lines  and  inter- 
secting each  other  at  the  intersection  of  the  curved  lines.  The 
limiting  positions  of  these  secants  will  be  tangent  lines  to  the 
cone,  and  as  these  tangents  intersect,  they  determine  a  tangent 
plane.  The  tangent  plane  thus  determined  is  identical  with 
that  obtained  from  the  limiting  position  of  a  secant  plane. 

Instead  of  drawing  two  intersecting  curves  on  the  surface 
of  the  cone,  it  is  possible  to  select  the  element  of  contact  and  any 
curve  on  the  cone  intersecting  it.  The  tangent  plane  in  this 
case  is  determined  by  the  element  of  contact  and  the  limiting 
position  of  one  secant  to  the  curve  through  the  intersection  of 
the  element  and  the  curve. 

As  another  example,  take  a  spherical  surface  and  on  it  draw 
two  intersecting  lines  (necessarily  curved).  Through  the  point 
of  intersection,  draw  two  secants,  one  to  each  curve  and  deter- 
mine the  tangent  positions.  Again,  the  plane  of  the  two  inter- 
secting tangents  is  the  tangent  to  the  sphere.  Hence,  as  a  general 
definition,  a  tangent  plane  is  the  plane  established  by  the  limiting 
position  of  two  intersecting  secants  as  the  points  of  secancy 
reach  coincidence. 

1018.  Normal  plane.     The  normal  plane  is  any  plane  that 
is  perpendicular  to  the  tangent  plane.     If  a  normal  plane  is  to 
be  drawn  to  a  sphere  at  a  given  point,  for  instance,  then  construct 
the  tangent  plane  and  draw  through  the  point  of  tangency  any 
plane  perpendicular  to  the  tangent  plane.     An  infinite  number 
of  normal  planes  may  be  drawn,  all  passing  through  the  given 


176  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

point.     The  various  normal  planes  will  intersect  in  a  common 
line,  which  is  normal  to  the  tangent  plane  at  the  point  of  tangency. 

1019.  Singly  curved   surface.     A  singly  curved   surface  is 
a  surface  whose  successive   rectilinear  elements  may  be   made 
to  coincide  with  a  plane.     Hence,  a  tangent  plane  must  be  in 
contact  all  along  some  one  rectilinear  element.     As  examples, 
the    conical,    cylindrical,    convolute    and    the    helicoidal    screw 
surfaces  may  be  mentioned. 

1020.  Doubly  curved  surface.     A  doubly  curved  surface  is 
a  surface  whose  tangent  plane  touches  its  surface  at  a  point. 
Evidently,  any  surface  which  is  not  plane  or  singly  curved  must 
be  doubly  curved.     The  sphere  is  a  familiar  example  of  a  doubly 
curved  surface. 

1021.  Singly  curved  surface  of  revolution.    A  singly  curved 
surface  of  revolution  is  a  surface  generated  by  a  straight  line 
revolving  about  another  straight  line  in  its  own  plane  as  an  axis, 
so  that  every  point  on  the  revolving  line  describes  a  circle  whose 
plane  is  perpendicular  to  the  axis,  and  whose  centre  is  in  the  axis. 
Thus,  only  two  cases  of  singly  curved  surfaces  "can  obtain,  the 
conical  and  the  cylindrical  surfaces  of  revolution. 

1022.  Doubly  curved  surface  of  revolution.     A   doubly 
curved  surface  of  revolution  is  a  surface  generated  by  a  plane 
curve  revolving  about  a  straight  line  in  its  own  plane  as  an  axis 
so  that  every  point  on  the  revolving  curve  describes  a  circle  whose 
plane  is  perpendicular  to  the  axis,  and  whose  centre  lies  in  the 
axis.     Hence,  there  are  infinite  varieties  of  doubly  curved  sur- 
faces of  revolution  as  the  sphere,   ellipsoid,   hyperboloid,  etc., 
generated  by  revolving  the  circle,  ellipse,  hyperbola,  etc.,  about 
their  axes.     In  the  case  of  the  parabola,  the  curve  may  revolve 
about  the  axis  or  the  directrix  in  which  cases  two  distinct  types 
of  surfaces  are  generated.     Similarly  with  the   hyperbola,  the 
curve  may  generate  the  hyperboloid  of  one  or  two  nappes  depend- 
ing upon  whether  the  conjugate  or  transverse  axis  is  the  axis  of 
revolution,  respectively. 

Sometimes,  a  distinction  is  made  between  the  outside  and 
inside  surfaces  of  a  doubly  curved  surface.  For  example,  the 
outside  surface  of  a  sphere  is  called  a  doubly  convex  surface, 
whereas,  the  inside  is  an  illustration  of  a  doubly  concave  surface. 


CLASSIFICATION  OF  SURFACES 


177 


A  circular  ring  made  of  round  wire,  and  known  as  a  torus,  is  an 
example  of  a  doubly  concavo-convex  surface. 

1023.  Revolution  of  a  skew  line.     An  interesting  surface 
is  the  one  generated  by  a  pair  of  skew 

lines  when  one  is  made  to  revolve  about 
the  other  as  an  axis.  Fig.  180  gives 
such  a  case,  and  as  no  plane  can  be 
passed  through  successive  elements,  it 
is  a  warped  surface.  While  revolving 
about  the  axis,  the  line  generates  the 
same  type  of  surface  as  would  be  gen- 
erated by  a  hyperbola  when  revolved 
about  its  conjugate  axis.  The  surface 
is  known  as  the  hyperboloid  of  revo- 
lution of  one  nappe,  and,  incidentally, 
is  the  only  warped  surface  of  revolu- 
tion. 

1024.  Meridian  plane  and   me- 
ridian  line.      If  a   plane  be  passed 
through  the  axis  of  a  doubly  curved 
surface  of  revolution,  it  will  cut  from 
the  surface  a   line  which  is  the  me- 
ridian  line.     The  plane   cutting  the  -pic  180 
meridian  line  is  called  the  meridian 

plane.  Any  meridian  line  can  be  used  as  the  generatrix  of  the  sur- 
face of  revolution,  because  all  meridian  lines  are  the  same.  The 
circle  is  the  meridian  line  of  a  sphere,  and  for  this  particular 
surface,  every  section  is  a  circle.  In  general, 
every  plane  perpendicular  to  the  axis  will  cut  a 
circle  from  any  surface  of  revolution,  whether 
singly  or  doubly  curved. 

1025.  Surfaces    of    revolution    having    a 
common  axis.     If  two  surfaces  of  revolution 
have  a  common  axis  and  the  surfaces  intersect, 
tangentially  or  angularly,  they  do  so  all  around 
in  a  circle  which  is  common  to  the  two  surfaces 
of  revolution.     Thus,  the  two  surfaces  shown  in 
Fig.  181  intersect,  angularly,  in  a  circle  having  ab  as  a  diameter, 
and  intersect,  tangentially,  in  a  circle  having  cd  as  a  diameter. 


FIG.  181. 


178 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


1026.  Representation  of  the  doubly  curved  surface 
of  revolution.  Fig.  182  shows  a  doubly  curved  surface  of 
revolution  shown  in  two  views.  One  view  shows  the  same  as 
that  produced  by  a  meridian  plane  cutting  a  meridian  line  and 
the  other  shows  the  same  as  concentric  circles.  When  considered 
as  a  solid,  no  ground  line  is  necessary  as  the  distance  from  the 
principal  planes  is  unimportant.  Centre  lines,  ab,  cd  and  ef 


FIG.  182. 


should  be  shown,  ab  and  ef  being  represented  as  two  lines,  be- 
cause both  views  are  distinct  from  each  other.  The  lines  indi- 
cate that  the  object  is  symmetrical  about  the  centre  line  as  an 


axis. 


1027.  To  assume  a  point  on  a  doubly  curved  surface  of 
revolution.     Let  c,  Fig.  183,  be  assumed  as  one  point  on  the 

surface.*  With  oc  as  a  radius, 
draw  the  arc  ca,  a  is,  there- 
fore, the  revolved  position  of 
c  when  the  meridian  plane 
through  co  has  been  revolved 
to  ao.  Hence,  a  is  at  a'  or 
a"  in  the  corresponding  view. 
On  counter-revolution,  a'  de- 
scribes a  circle,  the  plane  of  which  is  perpendicular  to  the  axis, 
and  the  plane  is  shown  by  its  trace  a'c';  in  the  other  view,  a 
returns  to  c  and,  hence,  c'  is  the  final  position.  It  may  also  be 
at  c"  for  the  same  reason,  but  then  c  is  hidden  in  that  view.  If, 
on  the  other  hand,  d  is  chosen  as  one  projection,  its  corre- 


FIG.  183. 


*  These  views  bear  third  angle  relation  to  each  other. 


CLASSIFICATION  OF  SURFACES  179 

spending  projection  is  d',  if  d  is  visible;   or,  it  is   d",  if  d  is 

hidden. 

1028.  Developable    surface.     When  a  curved  surface    can 
be  rolled  over  a  plane  surface  so  that  successive  elements  come 
in  contact  with  the  plane  and  that  the  area  of  the  curved  surface 
can  be  made  to  equal  the  plane  surface  by  rectification,  the 
surface  is  a  developable  one.     Hence,  any  singly  curved  surface, 
like  a  cylinder,  can  be  rolled  out  flat  or  developed.     A  sphere 
cannot  be  rolled  out  as  a  flat  surface  because  it  has  point  con- 
tact with  a  plane,  and  is,  therefore,  incapable  of  development. 
If  a  sphere  is  to  be  constructed  from  flat  sheets,  it  may  be  ap- 
proximated by  cutting  it  into  the  type  of  slices  called  lunes, 
resembling  very  much  the  slices  made  by  passing  meridian  planes 
through  the  axis  of  a  sphere.     To  approach  more  nearly  the 
sphere,  it  would  be  necessary  to  take  these  lunes  and  hammer 
them  so  as  to  stretch  the  material  to  the  proper  curvature.     Simi- 
larly, in  the  making  of  stove-pipe  elbows,  the  elbows  are  made 
of  limited  portions  of  cylinders  and  cut  to  a  wedge  shape  so  as 
to  approximate  the  doubly  curved  surface  known  as  the  torus. 

1029.  Ruled  surface.     Every  surface  on  which  a  straight 
line  may  be  drawn  is  called  a  ruled  surface.    A  ruled  surface 
may  be  plane,  singly  curved  or  doubly  curved.     Among  the  singly 
curved  examples  may  be  found  the  conical,  cylindrical,  convolute 
and  helicoidal  screw  surfaces.     The  hyperboloid  of  revolution 
of  one  nappe  furnishes  a  case  of  a  doubly  curved  ruled  surface 
(1023). 

1030.  Asymptotic  surface.     If  a  hyperbola  and  its  asymp- 
totes move  so  that  their  plane  continually  remains  parallel  to 
itself,  and  any  point  on  the  curve  or  on  the  asymptotes  touches 
a  straight  line  as  a  directrix,  the  hyperbola  will  generate  a  hy- 
perbolic  cylindrical  surface  and  the  asymptotes  will   generate 
a  pair  of  asymptotic  planes.    Also,  if  the  hyperbola  revolves 
about  the  transverse  or  conjugate  axis,  the  hyperbola  will  generate 
a  hyperboloid  of  revolution  and  the  asymptote  will  generate  a 
conical  surface  which  is  asymptotic  to  the  hyperboloid.     In  all 
cases,  the  asymptotic  surface  is  tangent  at  two  lines,  straight  or 
curved,  at  an  infinite  distance  apart  and  the  surface  passes  within 
finite  distance  of  the  intersection  of  the  axes  of  the  curve. 


180 


GEOMETRICAL  PROBLEMS   IN    PROJECTION 


1031.  Classification  of  surfaces. 


Surfaces. 


Ruled  surfaces. 
Straight  lines 
may  be  drawn  on 
the  resulting  sur- 
face. 


Singly  curved  sur- 
faces. May  be 
developed  into  a 
flat  surface  by  rec- 
tification. 


Doubly  curved 
surfaces.  Tan- 
gent plane 
touches  surface 
at  a  point. 


Planes.      Any    two 

points  when  joined 

by  a  straight  line  lie 

wholly   within   the 

surface. 

Conical  surfaces. 
Rectilinear  ele- 
ments pass  through 
a  given  point  in 
space  and  touch  a 
curved  directrix. 

Cylindrical  surfaces. 
Rectilinear  ele- 
ments are  parallel 
to  each  other  and 
touch  a  curved  di- 
rectrix. 

Convolute  surfaces. 
Rectilinear  ele- 
ments tangent  to  a 
line  of  double  curva- 
ture.    Consecutive 
elements    intersect 
two   and   two;    no 
three  intersect  in  a 
common  point. 
Warped   surfaces. 
No  two  consecutive 
elements  lie  in  the 
same  plane;  hence, 
they  are  non-devel- 
opable. 

Doubly    curved    surfaces    of    revolution. 

Generated  by  plane  curves  revolving  about 
an  axis  in  the  plane  of  the  curve.  All 
meridian  lines  equal  and  all  sections  per- 
pendicular to  axis  are  circles. 

Unclassified  doubly  curved  surfaces.  All 
others  which  do  not  fall  within  the  fore- 
going classification. 


QUESTIONS  ON  CHAPTER  X 


1.  How  are  surfaces  generated? 

2.  What  is  a  plane  surface? 

3.  What  is  a  rectilinear  generatrix? 

4.  WHat  is  a  directrix? 

5.  What  is  an  element  of  a  surface? 

6.  What  is  a  conical  surface? 


CLASSIFICATION  OF  SURFACES  181 

7.  What  is  the  directrix  of  a  conical  surface? 

8.  What  is  the  vertex  of  a  conical  surface? 

9.  Is  it  necessary  for  the  directrix  of  a  conical  surface  to  be  closed? 

10.  What  is  a  nappe  of  a  cone?    How  many  nappes  are  generated  in  a 

conical  surface? 

11.  What  is  a  cone? 

12.  What  is  the  base  of  a  cone? 

13.  Must  all  elements  be  cut  by  the  plane  of  the  base  for  a  cone? 

14.  What  is  a  circular  cone? 

15.  What  is  the  axis  of  a  cone? 

16.  What  is  a  right  circular  cone?     Is  it  a  cone  of  revolution? 

17.  What  is  the  altitude  of  a  cone? 

18.  What  is  the  slant  height  of  a  cone? 

19.  What  is  the  an  oblique  cone? 

20.  What  is  a  frustum  of  a  cone? 

21.  How  are  the  two  bases  of  a  frustum  of  a  cone  usually  designated? 

22.  What  is  a  truncated  cone? 

23.  In  the  representation  of  a  cone,  why  is  the  plane  of  the  base  usually 

assumed  perpendicular  to  the  plane  of  projection? 

24.  Is  it  necessary  that  the  base  of  a  cone  should  be  circular? 

25.  Draw  a  cone  in  orthographic  projection  and  assume  the  plane  of 

the  base  perpendicular  to  the  vertical  plane. 

26.  Draw  a  cone  in  projection  and  show  how  an  element  of  the  surface 

is  assumed  in  both  projections.    State  exactly  where  the  element 
is   chosen. 

27.  Draw  a  cone  in  projection  and  show  how  a  point  is  assumed  on  its 

surface.     Locate  point  in  both  projections. 

28.  What  is  a  cylindrical  surface? 

29.  Define  generatrix  of  a  cylindrical  surface;  directrix;  element. 

30.  Is  it  necessary  that  the  directrix  of  a  cylindrical  surface  be  a  closed 

curve? 

31.  How  is  a  cylinder  differentiated  from  a  cylindrical  surface? 

32.  How  many  bases  must  a  cylinder  have? 

33.  What  is  the  axis  of  a  cylinder? 

34.  Must  the  axis  of  the  cylinder  be  parallel  to  the  elements?    Why? 

35.  What  is  an  oblique  cylinder? 

36.  What  is  a  right  circular  cylinder?    Is  this  cylinder  a  cylinder  of 

revolution? 

37.  Is  it  necessary  that  a  right  cylinder  have  a  circle  for  the  base?    Why? 

38.  Draw  an  oblique  cylinder  whose  base  lies  in  the  horizontal  plane. 

39.  In  Question  38,  assume  an  element  of  the  surface  and  state  which 

element  is  chosen. 

40.  Which  elements  are  the  limiting  elements  in  Question  38? 

41.  Draw  a  cylinder  in  projection  and  then  assume  a  point  on  the  surface 

of  it.     Show  it  in  both  projections. 

42.  What  is  a  convolute  surface? 

43.  What  is  an  oblique  helicoidal  screw  surface?    Give  a  prominent 

example  of  it. 


182  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

44.  What  is  a  right  helicoidal  screw  surface?    Give  a  prominent  example 

of  it. 

45.  Show  that  the  helicoidal  screw  surfaces  are  limiting  helicoids  as  the 

inner  helix  becomes  of  zero  diameter. 

46.  What  is  a  warped  surface?    Illustrate  by  sticks. 

47.  What  is  a  secant  plane? 

48.  Show  how  a  tangent  plane  is  the  limiting  position  of  a  secant  plane 

to  a  conical  surface. 

49.  Show  how  a  tangent  plane  is  the  limiting  position  of  a  secant  plane 

to  a  cylindrical  surface. 

50.  Show  that  a  tangent  line  to  the  surface  is  the  limiting  position  of  a 

secant  drawn  to  a  curve  on  the  surface. 

51.  Show  how  two  intersecting  lines  may  be  drawn  on  a  curved  surface 

and  how  the  limiting  positions  of  two  secants  drawn  through 
this  point  of  intersection  determine  a  tangent  plane  to  the 
surface. 

52.  Show  how  one  element  of  a  conical  surface  and  one  limiting  posi- 

tion of  a  secant  determine  the  tangent  plane  to  the  conical 
surface. 

53.  Show  how  one  element  of  a  cylindrical  surface  and  one  limiting 

position  of  a  secant  determine  the  tangent  plane  to  the  cylindrical 
surface. 

54.  Show  how  two  intersecting  lines  may  be  drawn  on  the  surface  of  a 

sphere  and  how  the  limiting  positions  of  two  secants  drawn  through 
the  intersection  determine  a  tangent  plane  to  the  sphere. 

55.  Define  tangent  plane  in  terms  of  the  two  intersecting  tangent  lines 

at  a  point  on  a  surface. 

56.  Define  normal  plane. 

57.  How  many  normal  planes  may  be  drawn  through  a  given  point 

on  a  surface? 

58.  What  is  a  normal  (line)  to  a  surface? 

59.  Define  singly  curved  surface.     Give  examples. 

60.  Define  doubly  curved  surface.    Give  examples. 

61.  Define  singly  curved  surface  of  revolution.     Give  examples. 

62.  Define  doubly  curved  surface  of  revolution.     Give  examples. 

63.  What  is  a  doubly  convex  surface?    Give  examples. 

64.  What  is  a  doubly  concave  surface?    Give  examples. 

65.  What  is  a  doubly  concavo-convex  surface.     Give  examples. 

66.  Describe  the  surface  of  a  torus.     Is  this  a  concavo-convex  surface? 

67.  What  surface  is  obtained  when  a  pair  of  skew  lines  are  revolved 

about  one  of  them  as  an  axis? 

68.  Construct  the  surface  of  Question  67. 

69.  What  is  a  meridian  plane? 

70.  What  is  a  meridian  line? 

71.  Why  can  any  meridian  line  be  assumed  as  a  generatrix  for  its  partic- 

ular surface  of  revolution? 

72.  What  curves  are  obtained  by  passing  planes  perpendicular  to  the 

axis  of  revolution? 


CLASSIFICATION  OF  SURFACES  183 

73.  When  two  surfaces  of  revolution  have  the  same  axis,  show  that  the 

intersection  is  a  circle  whether  the  surfaces  intersect  tangentially 
or  angularly. 

74.  Show  how  a  doubly  curved  surface  of  revolution  may  be  represented 

without  the  ground  line.     Draw  the  proper  centre  lines. 

75.  Assume  a  point  on  the  surface  of  a  doubly  curved  surface  of  revolu- 

tion. 

76.  What  is  a  developable  surface? 

77.  Is  development  the  rectification  of  a  surface? 

78.  Are  singly  curved  surfaces  developable? 

79.  Are  doubly  curved  surfaces  developable? 

80.  Are  warped  surfaces  developable? 

81.  What  is  a  ruled  surface?     Give  examples. 

82.  What  is  an  asymptotic  plane?    Give  an  example. 

83.  If  a  hyperbola  and  its  asymptotes  revolve  about  the  transverse 

axis,  show  why  the  asymptotic  lines  generate  asymptotic  cones 
to  the  resulting  hyperboloids. 

84.  Show  what  changes  occur  in  Question  83  when  the  conjugate  axis 

is  used. 

85.  Make  a  classification  of  surfaces. 


CHAPTER  XI 


INTERSECTIONS  OF  SURFACES  BY  PLANES,  AND  THEIR 
DEVELOPMENT 

1101.  Introductory.  When  a  line  is  inclined  to  a  plane, 
it  will  if  sufficiently  produced,  pierce  the  plane  in  a  point. 
The  general  method  involved  has  been  shown  (Art.  823), 
for  straight  lines,  and  consists  of  passing  an  auxiliary  plane 
through  the  given  line,  so  that  it  cuts  the  given  plane  in  a  line 
of  intersection.  The  piercing  point  must  be  somewhere  on  this 
line  of  intersection  and  also  on  the  given  line;  hence  it  is  at  their 
intersection. 

In  the  case  of  doubly  curved  lines,  the  passing  of  auxiliary 
planes  through  them  is  evidently  impossible.  Curved  surfaces, 
instead  of  planes,  are  therefore  used  as  the  auxiliary  surfaces. 
Let,  for  example,  Fig.  184  show  a  doubly  curved  line,  and  let  the 

object  be  to  find  the  piercing 

point  of  the  doubly  curved  line 
on  the  principal  planes.  If  a 
cylindrical  surface  be  -passed 
through  the  given  line,  the 
elements  of  which  are  perpen- 
dicular to  the  horizontal  plane, 
it  will  have  the  curve  ab  as 
FIG.  184.  its  trace,  which  will  also  be  the 

horizontal     projection    of    the 

curve  AB  in  space.  Similarly,  the  vertical  projecting  cylindrical 
surface  will  cut  the  vertical  plane  in  the  line  a'b',  and  will  be  the 
vertical  projection  of  the  curve  AB  in  space.  If  a  perpendicular  be 
erected  at  a,  where  ab  crosses  the  ground  line,  it  will  intersect  the 
vertical  projection  of  the  curve  at  a',  the  vertical  piercing  point  of 
the  curve.  The  entire  process  in  substance  consists  of  this:  the 
surface  of  the  horizontal  projecting  cylinder  cuts  the  vertical 
plane  in  the  line  aa';  the  piercing  point  of  the  curve  AB  must 

184 


INTERSECTIONS  OF  SURFACES  BY  PLANES  185 

lie  on  aa'  and  also  on  AB,  hence  it  is  at  their  intersection  a'. 
Likewise,  b,  the  horizontal  piercing  point  is  found  by  a  process 
identical  with  that  immediately  preceding. 

1102.  Lines  of    intersection  of   solids  by  planes.     The 

extension  of  the  foregoing  is  the  entire  scheme  of  finding  the 
line  of  intersection  of  any  surface  with  the  cutting  plane.  Ele- 
ments of  the  surface  pierce  the  cutting  plane  in  points;  the  locus 
of  the  points  so  obtained,  determine  the  line  of  intersection. 

A  distinction  must  be  made  between  a  plane  cutting  a  sur- 
face, and  a  plane  cutting  a  solid.  In  the  former  case,  the  surface', 
alone,  gives  rise  to  the  line  of  intersection,  whether  it  be  an  open 
surface,  or  a  closed  surface;  the  cutting  plane  intersects  the 
surface  in  a  line  which  is  the  line  of  intersection.  In  the  latter 
case,  the  area  of  the  solid,  exposed  by  the  cutting  plane,  is  a 
section  of  the  solid.  This  is  the  scheme  of  using  section  planes 
for  the  elucidation  of  certain  views  in  drawing  (313). 

1103.  Development  of  surfaces.    The  development  of  a 
surface  consists  of  the  rolling  out  or  rectification  of  the  surface 
on  a  plane,  so  that  the  area  OH.  the  plane  is  equal  to  the  area 
of  the  surf  ace*  before  development.     If  this  flat  surface  be  rolled 
up,  it  will  regenerate  the  original  surface  from  which  it  has  been 
evolved  (1028). 

For  instance,  if  a  flat  rectangular  sheet  of  paper  be  rolled 
in  a  circular  form,  it  will  produce  the  surface  of  a  cylinder  of 
revolution.  Similarly,  a  sector  of  a  circle  may  be  wound  up 
so  as  to  make  a  right  circular  cone.  In  both  cases,  the  flat  sur- 
face is  the  development  of  the  surface  of  the  cylinder  or  cone 
as  the  case  may  be. 

1104.  Developable  surfaces.  A  prism  may  be  rolled  over 
a  flat  surface  and  each  face  successively  comes  in  intimate  con- 
tact with  the  flat  surface;  hence,  its  surface  is  developable.  A 
cylinder  may  likewise  be  rolled  over  a  flat  surface  and  the  con- 
secutive elements  will  successively  coincide  with  the  flat  sur- 
face, this,  again,  is  therefore  a  developable  surface.  The  sur- 
faces of  the  pyramid  and  the  cone  are  also  developable  for  similar 
reasons. 

A  sphere,  when  rolled  over  a  flat  surface,  touches  at  only 
one  point  and  not  along  any  one  element;  hence,  its  surface  is 


186  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

not  developable.  In  general,  any  surface  of  revolution  which 
has  a  curvilinear*  generatrix  is  non-developable. 

A  warped  surface  is  also  a  type  of  surface  which  is  non-develop- 
able, because  consecutive  elements  even  though  rectilinear, 
are  so  situated  that  no  plane  can  contain  them. 

Hence,  to  review,  only  singly  curved  surfaces,  and  such 
surfaces  as  are  made  up  of  intersecting  planes,  are  developable. 
Doubly  curved  surfaces  of  any  kind  are  only  developable  in  an 
approximate  way,  by  dividing  the  actual  surface  into  a  series 
of  developable  surfaces;  the  larger  the  series,  the  nearer  the 
approximation.! 

1105.  Problem  1.  To  find  the  line  of  intersection  of  the 
surfaces  of  a  right  octagonal  prism  with  a  plane  inclined  to  its 
axis. 

Construction.  Let  abed,  etc.,  Fig.  185,  be  the  plan  of  the 
prism,  resting,  for  convenience,  on  the  horizontal  plane.  The 
elevation  is  shown  by  a'b'c',  etc.  The  plane  T  is  perpendicular 
to  the  vertical  plane  and  makes  an  angle  a  with  the  horizontal 
plane.  In  the  supplementary  view  S,  only  half  of  the  intersec- 
tion is  shown,  because  the  other  half  is  symmetrical  about  the 
line  a"V".  Draw  from  e'd'c'b'  and  a',  lines  perpendicular 
to  Tt'  and  draw  a'"e"",  anywhere,  but  always  parallel  to  TV. 
The  axis  of  symmetry  ea,  shown  in  plan,  is  the  projection  of 
e'a'  in  elevation,  and  e'a'  is  equal  to  e'"a'";  the  points  e"'  and 
a'"  are  therefore  established  on  the  axis.  To  find  d'",  draw 
dm,  perpendicular  to  ea;  dm  is  shown  in  its  true  length  as  it  is 
parallel  to  the  horizontal  plane.  Accordingly,  set  off  m"'d'"  = 
md,  on  a  line  from  d',  perpendicular  to  Tt',  from  e'"a"'  as  a 
base  line.  As  nb  =  md,  then  n'"b'"=m'"d'",  and  is  set  off 
from  e"'a'",  on  a  line  from  V,  perpendicular  to  Tt'.  The  final 
point  c'"  is  located  so  that  o'"c"'  =  oc  and  is  set  off  from  e'"a'", 
on  a  line  from  c',  perpendicular  to  Tt.  The  "half  section"  is 

*  If  a  curvilinear  generatrix  moves  so  that  its  plane  remains  continually 
parallel  to  itself  and  touches  a  rectilinear  directrix,  the  surface  generated 
is  a  cylindrical  surface.  Hence,  this  surface  cannot  be  included  in  this 
connection. 

t  Maps  are  developments  of  the  earth's  surface,  made  in  various  ways. 
This  branch  of  the  subject  falls  under  Spherical  Projections.  The  student 
who  desires  to  pursue  this  branch  is  referred  to  the  treatises  on  Topographical 
Drawing  and  Surveying. 


INTERSECTIONS  OF  SURFACES  BY  PLANES 


187 


then  shown  completely  and  is  sectioned  as  is  customary.  The 
line  of  intersection  is  shown  by  *'"$"s'"\>'"9!" ;  the  half  section 
is  the  area  included  by  the  lines  e'''d'''c'"b'"a'"n'"o'"m'"e'". 

1106.  Problem  2.  To  find  the  developed  surfaces  in  the 
preceding  problem.* 

Construction.  As  the  prism  is  a  right  prism,  all  the  vertical 
edges  are  perpendicular  to  the  base;  the  base  will  develop  into  a 
straight  line  and  the  vertical  edges  will  be  perpendicular  to  it, 


FIG.  185. 

spaced  an  equal  distance  apart,  because  all  faces  are  equal. 
Hence,  on  the  base  line  AA,  Fig.  185,  lay  off  the  perimeter  of  the 
prism  and  divide  into  eight  equal  parts.  The  distance  of  a" 
above  the  base  line  AA  is  equal  to  the  distance  of  a'  above 
the  horizontal  plane;  b"  is  ablove  AA,  a  distance  equal  to  b' 
above  the  horizontal  plane.  In  this  way,  all  points  are  located. 
It  will  be  observed  that  the  development  is  symmetrical  about 
the  vertical  through  e". 

To  prove    the  accuracy  of  the    construction,  the  developed 
surface  may  be  laid  out  on  paper,  creased  along  all  the  verticals 

*  In  the  problems  to  follow  the  bases  are  not  included  in  the  development 
as  they  are  evident  from  the  drawing. 


188 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


and  wound  up  in  the  form  of  the  prism.  A  flat  card  will  admit 
of  being  placed  along  the  cut,  proving  that  the  section  is  a  plane 
surface. 

The  upper  and  lower  portions  are  both  developed;  the  dis- 
tance between  them  is  arbitrary,  only  the  cut  on  one  must  exactly 
match  with  the  cut  on  the  other. 

1107.  Problem  3.  To  find  the  line  of  intersection  of  the 
surface  of  a  right  circular  cylinder  with  a  plane  inclined  to  its 
axis. 

Construction.     Let  Fig.  186  show  the  cylinder  in  plan  and 


FIG.  186. 

elevation.  The  plane  cutting  it  is  an  angle  a  with  the  horizontal 
plane  and  is  perpendicular  to  the  vertical  plane.  It  is  customary, 
in  the  application,  to  select  the  position  of  the  plane  and  the 
object  so  that  subsequent  operations  become  most  convenient, 
so  long  as  the  given  conditions  are  satisfied.  Pass  a  series  of 
auxiliary  planes  through  the  axis  as  oe,  od,  etc.,  spaced,  for  con- 
venience, an  equal  angular  distance  apart.  These  planes  cut 
rectilinear  elements  from  the  cylinder,  shown  by  the  verticals 
through  e'd'c',  etc.  Lay  off  an  axis  e'"a'",  parallel  to  TV,  so  that 


e'"  e'  and  a'"a'  are  perpendicular  to  Tt'.     To   find   d' 


it   is 


INTERSECTIONS   OF  SURFACES  BY  PLANES  189 

known  that  md  is  shown  in  its  true  length  in  the  plan;  hence,  lay 
offm'"d'"=md  from  the  axis  e^a'",  on  a  line  from  d',  perpen- 
dicular to  Tt'.  Also,  make  o'"c'"  =  oc  and  n'"b"'  =  nb  in  the  same 
way.  Draw  a  smooth  curve  through  e^d"^'"^"^"^"^"^"^" 
which  will  be  found  to  be  an  ellipse.  The  ellipse  may  be  awn 
by  plotting  points  as  shown,  or,  the  major  axis  e'"a'"  and  the 
minor  axis  c'"c"'  may  be  laid  off  and  the  ellipse  drawn  by  any 
method.*  Both  methods  should  produce  identical  results. 
The  area  of  the  ellipse  is  the  section  of  the  cylinder  made  by 
an  oblique  plane  and  the  ellipse  is  the  curve  of  intersection. 
As  an  example,  a  cylindrical  glass  of  water  may  be  tilted  to  the 
given  angle,  and  the  boundary  of  the  surface  of  the  water  will 
be  elliptical. 

1108.  Problem  4.    To  find  the   developed  surface  in  the 
preceding  problem. 

Construction.  As  the  cylinder  is  a  right  cylinder,  the  ele- 
ments are  perpendicular  to  the  base,  and  the  base  will  develop 
into  a  straight  line,  of  a  length,  equal  to  the  rectified  length 
of  the  circular  base.  Divide  the  base  AA  Fig.  186,  into  the  same 
number  of  parts  as  are  cut  by  the  auxiliary  planes.  Every 
element  in  the  elevation  is  shown  in  its  true  size  because  it  is 
perpendicular  to  the  horizontal  plane;  hence,  make  a"  A  equal 
to  the  distance  of  a?  above  the  horizontal  plane;  b"A  equal 
to  the  distance  of  b'  above  the  horizontal  plane,  etc.  Draw  a 
smooth  curve  through  the  points  so  found  and  the  development 
will  appear  as  shown  at  D  in  Fig.  186.  Both  upper  and  lower 
portions  are  shown  developed,  either  one  may  be  wound  up  like 
the  original  cylinder  and  a  flat  card  placed 
across  the  intersection,  showing,  that  the  sur- 
face is  plane. 


1109.  Application  of  cylindrical  sur- 
faces. Fig.  187  shows  an  elbow,  approximat- 
ing a  torus,  made  of  sheet  metal,  by  the  use 
of  short  sections  of  cylinders.  The  ellipse  is 
symmetrical  about  both  axes,  and,  hence,  the  FIG.  187. 

upper  portion  of  the  cylinder  may  be  added  to 
the  lower  portion  so  as  to  give  an  offset.    Indeed  in  this  way  elbows 

*  See  Art.  906. 


190 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


FIG.  188. 


are  made  in  practice.  The  torus  is  a  doubly  curved  surface, 
and,  hence,  is  not  developable,  except  by  the  approximation 

shown.  To  develop  these  individual 
sheets,  pass  a  plane  ab,  perpendicular 
to  the  axis.  The  circle  cut  thereby 
will  then  develop  into  a  straight 
line.  Add  corresponding  distances 
above  and  below  the  base  line  so 
established  and  the  development 
may  be  completed.  Fig.  188  shows 
the  sheets  as  they  appear  in  the 
development. 

A    sufficient    number    of    points 
must  always  be  found  on  any  curve, 

so  that  no  doubt  occurs  as  to  its  form.  In  the  illustration,  the 
number  of  points  is  always  less  than  actually  required  so  as  to 
avoid  the  confusion  incident  to  a  large  number  of  construction  lines. 

1110.  Problem  5.  To  find  the  line  of  intersection  of  the 
surfaces  of  a  right  octagonal  pyramid  with  a  plane  inclined  to 
its  axis. 

Construction.  Let  Fig.  189  represent  the  pyramid  in  plan 
and  elevation,  and  let  T  be  the  cutting  plane.  The  plane  T 
cuts  the  extreme  edge  o'e'  at  e',  horizontally  projected  at  e, 
one  point  of  the  required  intersection.  It  also  cuts  the  edge 
o'd'  at  d',  horizontally  projected  at  d,  thus  locating  two  points 
on  the  required  intersection.  In  the  same  way  b  and  a  are 
located.  The  point  c  cannot  be  found  in  just  this  way.  If  the 
pyramid  be  turned  a  quarter  of  a  revolution,  the  point  c'  will  be 
at  q'  and  c'q'  will  be  the  distance  from  the  axis  where  the  edge 
oV  pierces  the  plane  T.  Hence,  lay  off  oc  =  c'q'  and  complete 
the  horizontal  projection  of  the  intersection. 

To  find  the  true  shape  of  the  intersection,  draw  the  supple- 
mentary view  S.  Lay  off  e"'a'"  as  the  axis,  parallel  to  Tt'; 
the  length  of  the  axis  is  such  that  e'a'  =  e'"a"'.  From  the 
horizontal  projection,  dm  is  made  equal  to  d'"m'"  in  the  supple- 
mentary view,  the  latter  being  set  off  from  the  axis  e'"a'",  and 
on  a  line  from  d',  perpendicular  to  Tt'.  Also,  co  =  c'"o'"  and 
bn  =  b'"n'".  Thus,  the  true  intersection  is  shown  in  the  supple- 
mentary view  S. 


INTERSECTIONS  OF  SURFACES  BY  PLANES 


191 


1111.  Problem  6.  To  find  the  developed  surfaces  in  the 
preceding  problem. 

Construction.  The  extreme  element  o's',  Fig.  189,  is 
shown  in  its  true  length  on  the  vertical  plane  because  it  is  parallel 
to  that  plane.  Accordingly,  with  any  point  o"  as  a  centre, 
draw  an  indefinite  arc  through  AA,  so  that  o"A  =  o's'.  Every 
edge  of  the  pyramid  meeting  at  the  vertex  has  the  same  length 
and  all  are  therefore  equal  to  o"A.  The  base  of  each  triangular 
face  is  shown  in  its  true  length  in  the  plan,  and,  hence,  with  any 
one  of  them  as  a  length,  set  the  distance  off  as  a  chord  on  AA 


.  FIG.  189. 

by  the  aid  of  a  divider,  so  that  the  number  of  steps  is  equal  to 
the  number  of  faces.  Draw  these  chords,  indicate  the  edges, 
and  it  then  remains  to  show  where  the  cutting  plane  intersects 
the  edges.  The  extreme  edge  oV  is  shown  in  its  true  length, 
hence,  lay  off  o"e"=o'e'.  The  edge  o'd'  is  not  shown  in  its 
true  length,  but  if  the  pyramid  is  revolved  so  that  this  edge 
reaches  the  position  o's',  then  d'  will  reach  p'  and  o'p'  is  the 
desired  true  length  as  it  is  now  parallel  to  the  vertical  plane; 
therefore,  set  off  o'p'  =  o"d".  In  this  way,  o'q'  =  o"c",  and  oY  = 
o"b".  The  edge  o'a'  is  shown  in  its  true  length,  therefore,  o'a' 
=  o"a";  and  a"  A  on  one  side  must  equal  a' 'A  on  the  other. 


192  GEOMETRICAL  PROBLEMS  IN   PROJECTION 

as,  on  rolling  up,  the  edges  must  correspond,  being  the  same 
initially.  Join  a"b",  b"c",  etc.,  to  complete  the  development. 
The  proof,  as  before,  lies  in  the  actual  construction  of  the  model 
and  in  showing  that  the  cut  is  a  plane  surface. 

1112.  Problem  7.  To  find  the  line  of  intersection  of  the 
surface  of  a  right  circular  cone  with  a  plane  inclined  to  its  axis. 

Construction.  Let  Fig.  190  show  plan  and  elevation  of  the 
cone,  and  let  T  be  the  cutting  plane.  Through  the  axis,  pass  a 


FIG.  190. 


series  of  planes  as  ao,  bo,  co,  etc.  These  planes  cut  rectilinear 
elements  from  the  cone,  shown  as  o'a',  o'b',  oV,  etc.,  in  the  eleva- 
tion. At  first,  it  is  a  good  plan  to  draw  the  horizontal  projection 
of  the  line  of  intersection,  as  many  points  are  readily  located 
thereon.  For  instance,  e'  is  the  point  where  the  extreme  ele- 
ment oV  pierces  the  plane  T,  horizontally  projected  at  e.  Sim- 
ilarly, the  actual  element  OD  pierces  the  plane  so  that  d  and  d' 
are  corresponding  projections.  The  points  b  and  b'  and  the 
points  a  and  a'  are  found  in  an  identical  manner,  but,  as  in  the 
case  of  the  pyramid,  the  point  c  cannot  be  located  in  the  same 
way.  If  the  cone  be  revolved  so  that  the  element  oV  occupies 


INTERSECTIONS   OF  SURFACES   BY  PLANES  193 

the  extreme  position  o's',  then  c'  will  be  found  at  q'  and  c'q' 
will  be  the  radius  of  the  circle  which  the  point  c'  describes;  hence, 
make  oc  =  c'q'  and  the  resultant  curve,  which  is  an  ellipse,  may 
be  drawn. 

The  true  line  of  intersection  is  shown  in  the  supplementary 
view  S.  As  in  previous  instances,  first  the  axis  e'"a"'  is  drawn 
parallel  to  TV,  and  equal  in  length  to  e'a'  so  that  e"'e'  and  a'"a'  are 
both  perpendicular  to  TV.  From  the  axis,  on  a  line  from  d', 
lay  off  d'"m'"  =  dm;  also  o'"  c'"=oc  and  b'"n'"=bn.  A  smooth 
curve  then  results  in  an  ellipse. 

If  accuracy  is  desired,  it  is  better  to  lay  off  the  major  and 
minor  axes  of  the  ellipse  and  then  draw  the  ellipse  by  other 
methods,*  since  great  care  must  be  used  in  this  construction. 
The  major  axis  is  shown  as  e'"a'".  To  find  the  minor  axis, 
bisect  e'a'  at  u'and  draw  uV,  the  trace  of  a  plane  perpendicular 
to  the  axis  of  the  cone.  This  plane  cuts  the  conical  surface  in  a 
circle,  of  which  wV  is  the  radius.  The  minor  axis  is  equal  to 
the  length  of  the  chord  of  a  circle  whose  radius  is  wV  and  whose 
distance  from  the  centre  of  the  circle  is  u'w'. 

1113.  Problem  8.  To  find  the  developed  surface  in  the 
preceding  problem. 

Construction.  The  extreme  visible  element  o's',  Fig.  190, 
is  shown  in  its  true  length  and  is  the  slant  height  of  the  cone. 
All  elements  are  of  the  same  length  and  hence,  any  indefinite 
arc  AA,  drawn  so  that  o"A  =  o's'  will  be  the  first  step  in  the 
development.  The  length  of  the  arc  AA  is  equal  to  the  rectified 
length  of  the  base,  and,  as  such,  is  laid  off  and  divided  into  any 
convenient  number  of  parts.  Eight  equal  parts  are  shown  here 
because  the  auxiliary  planes  were  chosen  so  as  to  cut  the  cone 
into  eight  equal  parts.  If  the  conical  surface  is  cut  along  the 
element  OA,  then  o'a'  .may  be  laid  off  on  both  sides  equal  to 
o"a"  in  the  development.  The  elements  o'd',  o'c'  and  o'b' 
are  not  shown  in  their  true  length,  therefore,  it  is  necessary  to 
revolve  the  cone  about  the  axis  so  as  to  make  them  parallel, 
in  turn,  to  the  vertical  plane.  The  points  will  ultimately  reach 
the  positions  indicated  by  p,'  q'  and  r',  and,  hence,  o"b"  =  o'r', 
o"c"  =  o'q'  and  o"d"  =  o'p'.  The  element  o'e'  is  shown  in  its 
true  length  and  is  laid  off  equal  to  o"e".  A  smooth  curve  through 
a"b"c"d",  etc.,  completes  the  required  development. 


'".etc.,  co 

*  See  Art.  906  in  this  connection. 


194 


GEOMETRICAL  PROBLEMS   IN   PROJECTION 


When  frustra  of  conical  surfaces  are  to  be  developed  the 
elements  of  the  surface  may  be  produced  until  they  meet  at  the 
vertex.      The    development  may  then    proceed 
along  the  usual  lines. 

1114.  Application     of    conical    surfaces. 

As  the  ellipse  is  a  symmetrical  curve,  the  upper 
and  lower  portions  of  the  cone  may  be  turned 
end  for  end  so  that  the  axes  intersect.  The 
resultant  shape  is  indicated  in  Fig.  191,  used  at 
times  in  various  sheet  metal  designs,  such  as 

oil-cans,  tea-kettles,  etc. 
FIG.  191. 

1115.  Problem  9.    To  find  the  line  of  inter- 
section of  a  doubly  curved  surface  of  revolution  with  a  plane 
inclined  to  its  axis. 

Construction.  Let  Fig.  192  show  the  elevation  of  the  given 
surface  of  revolution.*  Atten- 
tion will  be  directed  to  the 
construction  of  the  section 
shown  in  the  plan.  The  high- 
est point  on  the  curve  is  shown 
at  a,  which  is  found  from  its 
corresponding  projection  a',  at 
the  point  where  the  plane  T 
cuts  the  meridian  curve  that 
is  parallel  to  the  plane  of  the 
paper.  The  point  b  is  directly 
under  b'  and  the  length  bb  is 
equal  to  the  chord  of  a  circle 
whose  radius  is  m'n'  and 
whose  distance  from  the  cen- 
tre is  b'n'.  Similarly,  cc  is 
found  by  drawing  an  indefi- 
nite line  under  c'  intersected 
by  op  =  o'p'  as  a  radius.  As 

many  points  as  are  necessary  are  located,  so  as  to  get  the  true 
shape.     One  thing,  however,  must  be  observed:   The  plane  XT 

*  Many  of  these  constructions  can  be  carried  to  completion  without 
actually  showing  the  principal  planes.  The  operations  on  them  are  per- 
formed intuitively. 


FIG.  192. 


INTERSECTIONS  OF  SURFACES  BY  PLANES 


195 


cuts  the  base  of  the  surface  of  revolution  at  h'  in  the  elevation 
and  therefore  hh  is  a  straight  line,  which  is  the  chord  of  a  circle, 
whose  radius  is  q'r'  and  which  is  located  as  shown  in^the  plan. 

The  construction  of  the  supplementary  view  resembles,  in 
many  respects,  the  construction  in  plan.  The  similar  letters 
indicate  the  lengths  that  are  equal  to  each  other. 

1116.  Problem  10.  To  find  the  line  of  intersection  of  a  bell- 
surface  with  a  plane. 

Construction.  Fig.  193  shows  the  stub  end  of  a  connecting 
rod  as  used  on  a  steam  engine.  The  end  is  formed  in  the  lathe 
by  turning  a  bell-shaped  surface  of  revolution  on  a  bar  of  a 


rectangular  section,  as  shown  in  the  end  view,  the  shank  being 
circular.  The  radius  of  curvature  for  the  bell  surface  is  located 
on  the  line  ab  as  shown.  The  plane  TT  is  tangent  to  the  shank 
of  the  rod  and  begins  to  cut  the  bell-surface  at  points  to  the 
left  of  ab;  hence,  the  starting  point  of  the  curve  is  at  c.  To 
find  any  point  such  as  d,  for  instance,  pass  a  plane  through  d 
perpendicular  to  the  axis.  It  cuts  the  bell-surface  in  a  circle 
(1024)  whose  radius  is  od"  =  o'd'".  Where  the  circle  intersects 
the  plane  TT  at  d',  project  back  to  d,  which  is  the  required  point 
on  the  curve.  The  scheme  is  merely  this:  By  passing  auxiliary 
planes  perpendicular  to  the  axis,  circles  are  cut  from  the  bell- 
surface  which  pierce  the  bounding  planes  in  the  required  points 
of  intersection.  Attention  might  be  directed  to  the  piont  e 
which  is  on  both  plan  and  elevation.  This  is  true  because  the 
planes  TT  and  SS  intersect  in  a  line  which  can  pierce  the  bell- 


196 


GEOMETRICAL  PROBLEMS  IN   PROJECTION 


surface  at  only  one  point.  The  manner  in  which  nn  =  n'n'  is 
located  in  the  plan  will  be  seen  by  the  construction  lines  which 
are  included  in  the  illustration.  To  follow  the  description  is 
more  confusing  than  to  follow  the  drawing. 

1117.  Development  by  Triangulation.  In  the  developments 
so  far  considered,  only  the  cases  of  extreme  simplicity  were  selected. 
To  develop  the  surface  an  oblique  cone  or  an  oblique  cylinder 
requires  a  slightly  different  mode  of  procedure  than  was  used 


FIG.  194. 

heretofore.  If  the  surface  of  the  cone,  for  instance,  be  divided 
into  a  number  of  triangles  of  which  the  rectilinear  elements  form 
the  sides  and  the  rectified  base  forms  the  base,  it  is  possible  to 
plot  these  triangles,  one  by  one,  so  as  to  make  the  total  area 
of  the  triangles  equal  to  the  area  of  the  conical  surface  to  be 
developed.  The  method  is  simple  and  is  readily  applied  in 
practice.  Few  illustrations  will  make  matters  clearer. 

1118.  Problem  11.  To  develop  the  surfaces  of  an  oblique 
hexagonal  pyramid. 

Construction.  Fig.  194  shows  the  given  oblique  pyramid. 
It  is  first  necessary  to  find  the  true  lengths  of  all  the  elements  of 
the  surface.  In  the  case  of  the  pyramid,  the  edges  alone  need 


INTERSECTIONS  OF  SURFACES  BY  PLANES 


197 


be  considered.  Suppose  the  pyramid  is  rotated  about  a  perpen- 
dicular to  the  horizontal  plane,  through  o,  so  that  the  base  of 
the  pyramid  continually  remains  in  its  plane,  then,  when  oa 
is  parallel  to  the  vertical  plane,  it  is  projected  in  its  true  length 
and  is,  hence,  shown  as  the  line  o'a" '.  All  the  sides  are  thus 
brought  para  lei,  in  turn,  to  the  plane  and  the  true  lengths  are 
found.  In  most  cases  it  will  be  found  more  convenient  and  less 
confusing  to  make  a  separate  diagram  to  obtain  the  true  lengths. 
To  one  with  some  experience,  the  actual  lengths  need  not  be  drawn 
at  all,  but  simply  the  distances  a",  b",  c",  etc.,  laid  off.  Then 
from  any  point  o'",  draw  arcs  o'a"  =  o"'a'",  o'b"  =  o'"b'",  o'f"  = 


FIG.  195. 


o'"f",  etc.    From  a'", 


lay  off  distances  equal  to  the  respective 
in  this  case,  ab  =  bc  =  cd,  etc.,  and  hence  any  one  side 
will  answer  the  purpose.  Therefore,  take  that  length  on  a 
divider  and  step  off  ab  =  a''V"=b'/'c'"  =  c'"d"'  =  a''/f''',  etc. 
The  development  is  then  completed  by  drawing  the  proper  lines. 
The  case  selected,  shows  the  triangulation  method  applied 
to  a  surface  whose  sides  are  triangles.  It  is  extremely  simple 
for  that  reason,  but  its  simplicity  is  still  evident  in  the  case  of 
the  cone  as  will  be  now  shown. 

1119.  Problem  12.    To  develop  the  surface  of  an  oblique  cone. 

Construction.     Fig.  195  shows  the  given  oblique  cone.     The 

horizontal  projection  of  the  axis  is  op  and  the  base  is  a  circle, 


198 


GEOMETRICAL  PROBLEMS  IN    PROJECTION 


hence,  the  cone  is  not  a  cone  of  revolution  because  the  axis  is 
inclined  to  the  base.  Revolve  op  to  oq,  parallel  to  the  vertical 
plane.  Divide  the  base  into  any  number  of  parts  ab,  be,  cd, 
etc.  preferably  equal,  to  save  time  in  subsequent  operations. 
The  element  oa  is  parallel  to  the  plane,  hence,  o'a'  is  its  true 
length.  The  element  ob  is  not  parallel  to  the  plane,  but  can  be 
made  so  by  additional  revolution  as  shown;  hence  o'b'  is  its 
true  length.  From  any  point  o",  draw  o"a"  =  o'a',  o"b"  =  o'b', 
o"c"  =  o'c',  etc.  On  these  indefinite  arcs  step  off  distances  a"b"  = 
b"c"  =  c"d",  etc.,  equal  to  the  rectified  distances  ab  =  bc  =  cd, 
etc.  A  smooth  curve  through  the  points  a"b"c",  etc.,  will  give 


FIG.  195. 

the  development.  As  in  previous  problems,  the  cut  is  always 
made  so  as  to  make  the  shortest  seam  unless  other  requirements 
prevail. 

1120.  Problem  13.  To  develop  the  surface  of  an  oblique 
cylinder. 

Construction.  In  the  case  of  the  oblique  cone,  it  was  pos- 
sible to  divide  the  surface  into  a  series  of  triangles  which  were 
plotted,  one  by  one,  and  thus  the  development  followed  by  the 
addition  of  these  individual  triangles.  In  the  case  of  the  cylin- 
der, however,  the  application  is  somewhat  different,  although 
two  cones  may  be  used  each  of  which  has  one  base  of  the  cylinder 


INTERSECTIONS  OF  SURFACES  BY  PLANES 


199 


as  its  base.     The  application  of  this  is  cumbersome,  an  i  the 
better  plan  will  be  shown. 

Let  Fig.  196  show  the  oblique  cylinder,  chosen,  for  convenience, 
with  circular  bases.  Revolve  the  cylinder  as  shown,  until  it  is 
parallel  to  the  vertical  plane.  In  the  revolved  position,  ass  a 
plane  T  through  it,  perpendicular  to  the  axis.  The  curve  so  cut 
when  rectified,  will  develop  into  a  straight  line  and  the  elements 
of  the  surface  will  be  perpendicular  to  it.  The  true  shape  of  the 
section  of  the  cylinder  is  shown  by  the  curve  a"b"c"d",  etc., 
and  is  an  ellipse.  Its  construction  is  indicated  in  the  diagram. 
Draw  any  base  line  AA  and  lay  off  on  this  the  rectified  portions 


FIG.  196. 


of  the  ellipse  a"b"  =  a"'b'",  b"c"=b'"c'",  etc.  The  revolved 
positions  of  the  elements  shown  in  the  vertical  plane  are  all 
given  in  their  true  sizes  because  they  are  parallel  to  that  plane. 
Hence,  lay  off  a"'o'"  =  a'o',  b'"p'"  =  b'p',  etc.,  below  the  base 
line.  Do  the  same  for  the  elements  above  the  base  line  and  the 
curve  determined  by  the  points  so  found  will  be  the  development 
of  the  given  oblique  cylinder. 

1121.  Transition  pieces.  When  an  opening  of  one  cross- 
section  is  to  be  connected  with  an  opening  of  a  different  cross- 
section,  the  connecting  piece  is  called  a  transition  piece.  The 
case  of  transforming  a  circular  cross-section  to  a  square  cross- 
section  is  quite  common  in  heating  and  ventilating  flues,  boiler 
flues  and  the  like.  In  all  such  cases,  two  possible  methods  offer 


200 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


themselves  as  solutions.  The  two  surfaces  may  be  connected  by 
a  warped  surface,  the  rectilinear  elements  of  which  are  chosen  so 
that  the  corners  of  the  square  are  joined  with  the  quarter  points  on 
the  circle  while  the  intermediate  elements  are  distributed  between 
them.  A  warped  surface,  however,  is  not  developable,  it  is  also 
difficult  of  representation  and  therefore  commercially  unsuited 
for  application.  The  better  method  is  to  select  some  singly 
curved  surface,  or,  a  combination  of  planes  and  singly  curved 


FIG.  197. 

surfaces  since  these  are  developable.     The  application,   which 
is  quite  a  common  one,  is  shown  in  the  following  problem. 

1122.  Problem  14.  To  develop  the  surface  of  a  transition 
piece  connecting  a  circular  opening  with  a  square  opening. 

Construction.  Let  the  two  upper  views  of  Fig.  197  represent 
the  transition  piece  desired.  The  square  opening  is  indicated 
by  abed  and  the  circular  opening  by  efgh.  The  surface  may 
be  made  up  of  four  triangular  faces  aeb,  bfc,  cgd,  dha,  and  four 
conical  surfaces  hae,  ebf,  fcg,  gdh,  whose  vertices  are  at  a,  b,  c 
and  d. 

The  development  is  shown  at  D,  on  the  same  figure.     The 


INTERSECTIONS  OF  SURFACES  BY  PLANES 


201 


triangle  a"b"e"  is  first  laid  out  so  that  the  base  a"b"  =  ab; 
and  the  true  length  of  the  sides  are  determined  as  found  in  the 
construction  leading  to  the  position  a'e'  =  a"e"=b"e".  The 
conical  surfaces  are  developed  by  triangulation.  For  example, 
the  arc  he  is  divided  into  any  number  of  parts  em,  mn,  etc.; 
and  the  true  lengths  of  the  elements  of  the  surface  are  found 


FIG.  198. 


by  the  construction  leading  up  to  the  positions  a'e',  a'm',  a'n'. 
Hence,  a"e"=a'e',  a"m"  =  a'm' ;  and  m"e"=me  is  the  rectified 
arc  of  the  base  of  the  conical  surface.  When  four  of  these  com- 
binations of  triangle  and  conical  surface  are  laid  out,  in  their 
proper  order,  the  development  is  then  complete. 

If  a  rectangular  opening  is  to  be  joined  to  an  elliptical  open- 
ing, the  manner  in  which 
this  may  be  accomplished 
is  shown  in  Fig.  198. 

1123.  Problem  15. 
To  develop  the  surface  a 
transition  piece  connect- 
ing two  elliptical  openings 
whose  major  axes  are  at 
right  angles  to  each  other. 

Construction.  Fig. 
199  shows  three  views  of 
the  elliptical  transition 
piece.  By  reference  to 
the  diagram,  it  will  be 
seen  that  the  surface 
may  be  divided  into 

eight  conical  surfaces,  turned  end  for  end;  that  is,  four  vertices 
are  situated  on  one  ellipse  and  four  vertices  are  situated  on  the 


FIG.  199. 


202 


GEOMETRICAL  PROBLEMS   IN   PROJECTION 


other. 


The    shaded   figure   indicates   the  arrangement   of   the 

conical  surfaces  and  the 
a/  elements    are    shown    as 
shade  lines. 

In  all  developments  of 
this  character,  the  ar- 
rangement of  the  surfaces 
has  considerable  effect  on 
the  appearance  of  the 
transition  piece  when 
completed.  In  the  case 
chosen,  the  eight  conical 
surfaces  give  a  pleasing 
result.  If  the  intersec- 
tion of  the  eight  alter- 


FIG.  199. 


nate    conical    surfaces  is 
objectionable    then  it  is 

possible  to  use  a  still  larger  number  of  divisions. 

The  development  of  the  conical  surfaces  is  performed  by  the 

triangulation  method.     When  finished,   its   appearance   is  that 


/  \/\     / 


b'"  V"  b'» 

FIG.  200. 


b"f 


of  Fig.  200.     The  similar  letters  indicate  the  order  in  which  the 
surfaces  are  assembled. 

1124.  Development  of  doubly  curved  surfaces  by  approx=» 
imation.  Doubly  curved  surfaces  are  non-developable  because 
successive  elements  cannot  be  made  to  coincide  with  a  plane.  It 
is  possible,  however,  to  divide  the  surface  into  a  series  of  singly 
curved  surfaces  and  then  to  develop  these.  By  dividing  the 
doubly  curved  surface  into  a  sufficiently  large  number  of  parts, 


INTEKSECTIONS  OF  SURFACES  BY  PLANES 


203 


the  approximation  may  be  made  to  approach  the  surface  as 
closely  as  desired. 

For  doubly  curved  surfaces  of  revolution,  two  general  methods 
are  used.  One  method  is  to  pass  a  series  of  meridian  planes 
through  the  axis  and  then  to  adopt  a  singly  curved  surface 
whose  contour  is  that  of  the  surface  of  revolution  at  the 
meridian  planes.  This  method  is  known  as  the  gore  method. 

The  second  method,  known  as  the  zone  method,  is  to  divide 
the  surface  into  frusta  of  cones  whose  vertices  lie  on  the  axis 
of  the  doubly  curved  surface  of  revolution.  The  following 
examples  will  illustrate  the  application  of  the  methods.  . 

1125.  Problem  16.  To  develop  the  surface  of  a  sphere  by 
the  gore  method. 

Construction.     Let  the  plan  P  of  Fig.  201  show  the  sphere. 


FIG.  201. 

Pass  a  series  of  meridian  planes  aa,  bb,  cc,  dd,  through  the  sphere 
and  then  join  ab,  be,  cd,  etc.  From  the  plan  P  construct  the 
elevation  E  and  then  pass  planes  through  i'  and  g',  perpendicular 
to  the  axis  of  the  sphere,  intercepting  equal  arcs  on  the  meridian 
circle,  for  convenience.  Find  the  corresponding  circles  in  the 
plan  and  inscribe  an  octagon  (for  this  case)  and  determine  ef, 
the  chord  length  for  that  position  of  the  cutting  planes  through 
i'  and  g'.  This  distance  may  then  be  laid  off  as  e'f  in  the  eleva- 
tion E  and  the  curve  n'e'a'e'm'  be  drawn. 

For  the  development  D,  it  will  be  noted  that  ab  =  a'b'  =  a"b" 


204 


GEOMETRICAL  PROBLEMS   IN   PROJECTION 


m"g", 


and  ef  =  e'f'  =  e"f".  Also,  the  rectified  distance  m'g' 
g'h'=g"h",  h'i'=h"i",  etc.  A  suitable  number  of  points  must 
be  used  in  order  to  insure  the  proper  degree  of  accuracy,  the 
number  chosen  here  is  insufficient  for  practical  purposes.  By 
adding  eight  of  these  faces  along  the  line  a"b",  the  development 
is  completed. 

When  this  method  is  commercially  applied,  the  gores  may  be 
stretched  by  hammering  or  pressing  to  their  true  shape.  In 
this  case  they  are  singly  curved  surfaces  no  longer. 


FIG.  201. 

1126.  Problem  17.  To  develop  the  surface  of  a  sphere  by 
the  zone  method. 

Construction.  Let  P,  Fig.  202,  be  the  plan  and  E,  the  eleva- 
tion of  a  sphere.  Pass  a  series  of  planes  a'a',  b'b',  cV  etc., 
perpendicular  to  the  axis  of  the  sphere.  Join  a'b',  and  pro- 
duce to  h',  the  vertex  of  a  conical  surface  giving  rise  to  the  frustum 
a'b'b'a'.  Similarly,  join  bV  and  produce  to  g'.  The  last  conical 
surface  has  its  vertex  e'  in  the  circumscribing  sphere. 

The  development  D  is  s  milar  to  the  developnent  of  any 
frustum  of  a  conical  surface  of  revolution.  That  is,  with  h" 
as  a  centre,  draw  an  arc  with  h"a"=h'a'  as  a  radius.  At  the 
centre  line  make  a"o"  =  ao,  o"n"  =  on  etc.  The  radius  h"b"  =  h'b' 
so  that  the  arc  b"b"b"  is  tangent  internally,  to  the  similarly 
lettered  arc.  The  radius  for  this  case  is  g"b"=g'=b'.  The 
development  is  completed  by  continuing  this  process  until  all 
the  conical  surfaces  are  developed. 


INTERSECTIONS  OF  SURFACES  BY  PLANES 

h" 


205 


FIG.  202. 


1127.  Problem  18.  To  develop  a  doubly  curved  surface  of 
revolution  by  the  gore  method.  Let 
Fig.  203  represent  the  doubly  curved 
surface  of  revolution  as  finally  approxi- 
mated. Through  the  axis  at  h  pass  a 
series  of  equally  spaced  meridian 
planes,  and  then  draw  the  chords,  one 
of  which  is  lettered  as  dd.  In  the  ele- 
vation E,  pass  a  series  of  planes  k'k', 
IT,  m'm',  perpendicular  to  the  axis. 
The  lines  b'b',  cV,  d'd',  etc.,  are 
drawn  on  the  resulting  singly  curved 
surface  and  are  shown  in  their  true 
length.  The  corresponding  projections 
of  these  lines  in  the  plan  P  are  also 
shown  in  their  true  length. 


FIG.  203. 
To  develop  the  surface  of  one  face  lay  off  the  rectified  dis- 


206 


GEOMETRICAL  PROBLEMS  IN    PROJECTION 


tance   j'k'  =  j" 


FlQ  203 


,  etc.  At  the  points  j",  k",  1", 
etc.,  lay  off  a'a'  =  a"a",  b'b'  =  b"b", 
etc.,  perpendicular  to  h"j"  so  that 
the  resulting  figure  D  is  symmetrical 
about  it.  When  eight  of  these  faces 
(or  gores)  are  joined  together  and 
secured  along  the  seams,  the  resulting 
figure  will  be  that  shown  in  plan  and 
elevation. 

This  surface  may  also  be  approxi- 
„  mated   by   the    zone    method.      The 
~//  choice  of  method  is  usually  governed 
by    commercial    considerations.     The 
gore    method    is    perhaps    the    more 
economical  in  material. 


QUESTIONS  ON  CHAPTER  XI 

1.  State  the  general  method  of  finding  the  piercing  point  of  a  line  on 

a  plane. 

2.  Show  by  an  oblique  projection  how  the  piercing  points  of  a  doubly 

curved  line  are  found  on  the  principal  planes. 

3.  Why  is  the  projecting  surface  of  a  doubly  curved  line  a  projecting 

cylindrical  surface? 

4.  What  is  a  "line  of  intersection"  of  two  surfaces? 

5.  Distinguish  between  "line  of  intersection"  and  "section"  of  a  solid. 

6.  What  is  meant  by  the  development  of  a  surface?    Explain  fully. 

7.  What  are  the  essential  characteristics  of  a  developable  surface? 

8.  Why  is  a  sphere  non-developable? 

9.  Why  is  any  doubly  curved  surface  non-developable? 

10.  Why  is  a  warped  surface  non-developable? 

11.  Prove  the  general  case  of  finding  the  intersection  of  a  right  prism 

and  a  plane  inclined  to  its  axis. 

12.  Show  the  general  method  of  finding  the  development  of  the  prism  in 

Question  11. 

13.  Prove  the  general  case  of  finding  the  intersection  of  a  right  circular 

cylinder  and  a  plane  inclined  to  its  axis. 

14.  Show  the  general  method  of  finding  the  development  of  the  cylinder 

in  Question  13. 

15.  When  a  right  circular  cylinder  is  cut  by  a  plane  why  must  the  ellipse 

be  reversible? 

16.  How  is  the  torus  approximated  with  cylindrical  surfaces? 

17.  Prove  the  general  case  of  finding  the  intersection  of  a  right  pyramid 

and  a  plane  inclined  to  its  axis. 


INTERSECTIONS  OF  SURFACES  BY  PLANES  207 

18.  Show  the  general  method  of  finding  the  development  of  the  right 

pyramid  in  Question  17. 

19.  Prove  the  general  case  of  finding  the  intersection  of  a  right  circular 

cone  and  a  plane  inclined  to  its  axis. 

20.  Show  the  general  method  of  finding  the  development  of  the  right 

cone  in  Question  19. 

21.  Why  is  the  ellipse,  cut  from  a  cone  of  revolution  by  an  inclined  plane, 

reversible? 

22.  Prove  the  general  case  of  finding  the  intersection  of  a  doubly  curved 

surface  of  revolution  and  a  plane  inclined  to  the  axis. 

23.  Is  the  surface  in  Question  22  developable?    Why? 

24.  Prove  the  general  case  of  finding  the  intersection  of  a  bell-surface 

with  planes  parallel  to  its  axis. 

25.  Is  the  bell-surface  developable?     Why? 

26.  What  is  development  by  triangulation? 

27.  Prove  the  general  case  of  the  development  of  the  surface  of  an  oblique 

pyramid. 

28.  Prove  the  general  case  of  the  development  of  the  surface  of  an  oblique 

cone. 

29.  Prove  the  general  case  of  the  development  of  the  surface  of  an  oblique 

cylinder. 

30.  What  is  a  transition  piece? 

31.  Why  is  it  desirable  to  divide  the  surfaces  of  a  transition  piece  into 

developable  surfaces? 

32.  Prove  the  general  case  of  the  development  of  a  transition  piece  which 

joins  a  circular  opening  with  a  square  opening. 

33.  Prove  the  general  case  of  the  development  of  a  transition  piece 

which  joins  an  elliptical  opening  with  a  rectangular  opening. 

34.  Prove  the  general  case  of  the  development  of  a  transition  piece 

which  joins  two  elliptical  openings  whose  major  axes  are  at  right 
angles  to  each  other. 

35.  What  is  the  gore  method  of  developing  doubly  curved  surfaces? 

36.  What  is  the  zone  method  of  developing  doubly  curved  surfaces? 

37.  Prove  the  general  case  of  the  development  of  a  sphere  by  the  gore 

method. 

38.  Prove  the  general  case  of  the  development  of  a  sphere  by  the  zone 

method. 

39.  Prove  the  general  case  of  the  development  of  a  doubly  curved  sur- 

face of  revolution  by  the  gore  method. 

40.  A  right   octagonal   prism   has   a   circumscribing  circle  of  2"   and 

is  4"  high.  It  is  cut  by  a  plane,  inclined  30°  to  the  axis, 
and  passes  through  its  axis,  If"  from  the  base.  Find  the 
section. 

41.  Develop  the  surface  of  the  prism  of  Question  40. 

42.  A  cylinder  of  revolution  is  1\"  in  diameter  and  3f "  high.     It  is  cut 

by  a  plane,  inclined  45°  to  the  axis  and  passes  through  its  axis 
2"  from  the  base.  Find  the  section. 

43.  Develop  the  surface  of  the  cylinder  of  Question  42. 


208  GEOMETRICAL  PROBLEMS    IN   PROJECTION 

44.  A  right  octagonal  pyramid  has  a  circumscribing  base  circle  of  2\" 

and  is  4"  high.  It  is  cut  by  a  plane,  inclined  30°  to  the  axis  and 
passes  through  its  axis  2"  from  the  base.  Find  the  section. 

45.  Develop  the  surface  of  the  pyramid  of  Question  44. 

46.  A  right  circular  cone  has  a  base  2"  in  diameter  and  is  3f"  high. 

It  is  cut  by  plane  inclined  45°  to  the  axis  and  passes  through  its 
axis  21"  from  the  base.  Find  the  section. 

47.  Develop  the  cone  of  Question  46. 

48.  A  90°  stove  pipe  elbow  is  to  be  made  from  cylinders  4"  in  diameter. 

The  radius  of  the  -bend  is  to  be  18".  Divide  elbow  into  8  parts. 
The  tangent  distance  beyond  the  quadrant  is  4".  Develop  the 
surface  to  suitable  scale. 

49.  Assume  a  design  of  a  vase  (a  doubly  curved  surface  of  revolution) 

and  make  a  section  of  it. 

50.  The  stub  end  of  a  connecting  rod  of  a  steam  engine  is  4"x6*';  the 

rod  diameter  is  3£".  The  bell-surface  has  a  radius  of  12".  Find 
the  lines  of  intersection.  Draw  to  suitable  scale. 

51.  An  oblique  pyramid  has  a  regular  hexagon  for  a  base.    The  circum- 

scribing circle  for  the  base  has  a  diameter  of  2";  the  altitude  is 
3£"  and  the  projection  of  the  apex  is  2|"  from  the  centre  of  the 
base.  Develop  the  surface  of  the  pyramid. 

52.  An  oblique  cone  has  a  circular  base  of  2|"  in  diameter  and  an  altitude 

of  3".  The  projection  of  the  vertex  on  the  plane  of  the  base  is 
li"  from  the  centre  of  the  base.  Develop  the  conical  surface. 

53.  An  oblique  cylinder  has  a  circular  base  If"  in  diameter;  it  is  2£" 

high  and  the  inclination  of  the  axis  is  30°  with  the  base.  Develop 
the  surface. 

54.  A  transition  piece  is  to  be  made  joining  a  3'-0"x5'-6"  opening 

with  a  4'-0"  diameter  circle.  The  distance  between  openings 
3'-6".  Develop  the  surface.  Use  a  suitable  scale  for  the  drawing. 

55.  A  transition  piece  is  to  be  made  joining  an  opening  having  a  3'-0" 

X4'-0"  opening  to  a  2'-0"x6/-0"  opening  (both  rectangular). 
The  distance  between  openings  is  4'-0".  Develop  the  sheet  and 
use  suitable  scale  for  the  drawing. 

56.  A  square  opening  4"-0"x4'-0"  is  to  be  joined  with  an  elliptical 

opening  whose  major  and  minor  axes  are  5'-0"  X  3'  —0"  respec- 
tively. The  distance  between  openings  is  3'-0".  Develop  the 
sheet.  Use  suitable  scale  in  making  the  drawing. 

57.  A  circular  opening  having  a  diameter  of  3'-0"  is  to  be  connected 

with  an  elliptical  opening  whose  major  and  minor  axes  are 
4'-0"  x2'-6"  respectively.  The  distance  between  openings  is 
6/-0".  Develop  the  sheet.  Use  suitable  scale  in  making  the 
drawing. 

58.  An  opening  having  two  parallel  sides  and  two  semicircular  ends  has 

overall  dimensions  of  3'-0"  X5'-6".  This  opening  is  to  be  joined 
with  a  similar  opening  having  dimensions  of  3'-6"x4'-6".  The 
distance  between  openings  is  3'-6".  Develop  the  sheet.  Use 
suitable  scale  in  making  the  drawing. 


INTERSECTIONS  OF  SURFACES  BY  PLANES  209 

59.  A  frustum  of  a  cone  has  a  base  which  is  made  up  of  two  parallel 

sides  and  two  semicircular  ends  and  whose  overall  dimensions  are 
3'-6"  X6'-0".  The  height  of  the  vertex  above  the  base  is  8'-6"; 
that  of  the  upper  base  is  4'-0"  above  the  lower  base.  Develop 
the  sheet  and  use  suitable  scale  in  making  the  drawing. 

60.  An  elliptical  opening  having  a  major  and  minor  axis  of  5'-0"  and 

4'-0",  respectively,  is  to  be  joined  with  a  similar  opening  but 
whose  major  axis  is  at  a  right  angle.  Develop  the  sheet  and  use 
suitable  scale  in  making  the  drawing. 

61.  An  elliptical  opening  having  a  major  and  minor  axis  of  4/-6//  and 

3'-6",  respectively,  is  to  be  joined  with  a  similar  opening  but 
whose  major  axis  is  at  an  angle  of  30°.  Develop  the  sheet  and  use 
suitable  scale  in  making  the  drawing. 

62.  An  elliptical  opening  having  a  major  and  minor  axis  of  5'-0"  and 

3'-6",  respectively,  is  to  be  joined  with  another  elliptical  opening 
whose  major  and  minor  axes  are  4/-6//  and  3'-6",  respectively. 
Develop  the  sheet  and  use  suitable  scale  in  making  the  drawing. 

63.  A  sphere  is  6"  in  diameter.     Develop   the   surface   by   the   gore 

method  and  divide  the  entire  surface  into  sixteen  parts.  Use  a 
suitable  scale  in  making  the  drawing. 

64.  A  sphere  is  6"  in  diameter.     Develop  the  surface  by  the  zone 

method  and  divide  the  entire  surface  into  twelve  parts.  Use  a 
suitable  scale  in  making  the  drawing. 

65.  Assume  some  doubly  curved  surface  of  revolution  and  develop  the 

surface  by  the  gore  method. 

66.  Assume  some  doubly  curved  surface  of  revolution  and  develop  the 

surface  by  the  zone  method. 


CHAPTER  XII 

INTERSECTIONS  OF  SURFACES  WITH  EACH  OTHER,  AND  THEIR 

DEVELOPMENT 

1201.  Introductory.  When  two  surfaces  are  so  situated 
with  respect  to  each  other  that  they  intersect,  they  do  so  in  a 
line  which  is  called  the  line  of  intersection.  It  is  highly  desirable 
in  the  conception  of  intersections  to  realize  that  certain  elements 
of  one  surface  intersect  certain  elements  on  the  other  surface, 
and  that  the  locus  of  these  intersecting  elements  is  the  desired 
line  of  intersection. 

The  line  of  intersection  is  found  by  passing  auxiliary  surfaces 
through  the  given  surfaces.  Lines  will  thus  be  cut  from  the  given 
surfaces  by  the  auxiliary  surfaces,  the  intersections  of  \vhich 
yield  points  on  the  desired  line  of  intersection.  The  simplest 
auxiliary  surface  is  naturally  the  plane,  but  sometimes  cylindrical,* 
conical  and  spherical  surfaces  may  find  application.  It  is  not 
necessary  only  to  use  a  simple  type  of  auxiliary  surface,  but 
also  to  make  the  surface  pass  through  the  given  surfaces  so  as 
to  cut  them  in  easily  determinable  lines — preferably,  in  the 
elements.  As  illustrations  of  the  latter,  the  case  of  a  concial 
surface  cut  by  a  plane  passing  through  the  vertex  furnishes  an 
example.  It  cuts  the  surface  in  straight  lines  or  rectilinear 
elements.  Likewise,  a  plane  passed  through  a  cylindrical  sur- 
face parallel  to  any  element  will  cut  from  it,  one  or  more  lines 
which  are  also  rectiliner  elements.  For  doubly  curved  surfaces 
of  revolution,  a  plane  passed  through  the  axis  (a  meridian  plane) 
cuts  it  in  a  meridian  curve.  When  the  plane  is  perpendicular 
to  the  axis,  it  cuts  it  in  circles.  An  interesting  and  valuable 

*  A  careful  distinction  must  be  made  between  solids  and  their  bounding 
surfaces.  The  terms,  cylinder  and  cylindrical  surface  are  often  used  indis- 
criminately. The  associated  idea  is  usually  obtained  from  the  nature  of  the 
problem. 

210 


INTERSECTIONS  OF  SURFACES  WITH  EACH  OTHER    211 


property  of  spheres  is  that  all  plane  sections  in  any  direction, 
or  in  any  place,  are  all  circles,  whose  radii  are  readily  obtain- 
able. 

When  prisms  or  pyramids  intersect,  auxiliary  planes  are 
not  required  to  find  their  intersection,  because  the  plane  faces  and 
the  edges  furnish  sufficient  material  with  which  to  accomplish 
the  desired  result.  These  types  of  surfaces  are  therefore  exempt 
from  the  foregoing  general  method. 

1202.  Problem   1.    To  find  the  line  of  intersection  o*  the 
surfaces  of  two  prisms. 

Construction.  Let  Fig.  204  represent  the  two  prisms,  one 
of  which,  for  convenience,  is  a  right 
hexagonal  prism.  The  face  ABDC 
intersects  the  right  prism  in  the  line 
mn,  one  line  of  the  required  intersec- 
tion. The  face  CDFE  is  intersected 
by  the  edge  of  the  right  prism  in  the 
point  o,  found  as  shown  in  the  con- 
struction. Hence,  mo  is  the  next  line 
of  the  required  intersection.  Similarly, 
the  edge  EF  pierces  the  faces  of  the 
right  prism  at  p,  and  op  is  the  con- 
tinuation of  the  intersection.  In  this 
way,  all  points  are  found.  The  curve 
must  be  closed  because  the  inter- 
penetration  is  complete.  Only  the 

one  end  where  the  prism  enters  is  shown  in  construction. 
The  curve  where  it  again  emerges  is  found  in  an  identical 
manner. 

1203.  Problem  2.    To  find  the  developments  in  the  preceding 
problem. 

Construction.  The  development  of  the  right  prism  is 
quite  simple  and  should  be  understood  from  the  preceding 
chapter.  The  points  at  which  the  oblique  prism  inter- 
sects the  right  prism  are  also  shown.  Thus,  it  is  only 
necessary  to  make  sure  on  what  element  the  point  pierces, 
and  this  may  be  obtained  from  the  plan  view  of  the  right 
prism. 


O,k  c  i  eO 


FIG.  204. 


212 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


In  developing  the  oblique  prism,  D'  of  Fig.  205,  the  same 
general  scheme  is  followed  as  was  used  in  developing  the  oblique 
cylinder  (1120).  Revolve  the  prism  of  Fig.  204  so  that  the 
elements  are  parallel  to  the  plane  of  projection;  they  are  then 
shown  in  their  true  length.  Pass  a  plane  perpendicular  to  the 
edges  and  this  line  of  intersection  will  develop  into  a  straight  line 
which  is  used  as  a  base  line.  Find  now  the  true  section  of  the 
prism  and  lay  off  the  sides  perpendicular  to  the  base  line  and  at 
their  proper  distances  apart.  Lay  off  the  lengths  of  the  edges 
above  and  below  the  base  line  and  join  the  points  by  lines  to 
complete  the  development. 

When  actually  constructing  this  as  a  model,  it  will  be  well  to 


D 


D 


FIG.  205. 

carry  out  the  work  as  shown,  and  make  the  oblique  prism  in 
one  piece.  It  may  then  be  inserted  in  the  opening  provided  for 
in  the  right  prism,  as  indicated. 

1204.  Problem  3.  To  find  the  line  of  intersection  of  two  cylin- 
drical surfaces  of  revolution  whose  axes  intersect  at  a  right  angle. 

Construction.  Let  Fig.  206,  represent  the  given  cylindrical 
surfaces.  Through  o',  pass  a  series  of  planes  a' a',  b'b7,  cV, 
etc.  The  elements  cut  from  the  cylindrical  surface  by  these 
planes  interest  the  other  cylindrical  surface  at  the  points  a,  e,  b,  d,  c. 
The  two  upper  views  lead  to  the  construct  on  of  the  lower  view. 
It  will  be  seen  by  this  construction  that  the  entire  scheme  of 
locating  points  on  the  curve  lies  in  the  finding  of  the  successive 
intersections  of  the  elements  of  the  surfaces. 

If  the  cylindrical  surfaces  have  elliptical  sections,  instead  of 


INTERSECTIONS  OF  SURFACES  WITH  EACH  OTHER    213 


circular  sections  as  shown,  the  method  of  procedure  will  be  found 
to  be  much  the  same. 

1205.  Problem  4.  To  find  the  developments  in  the  preceding 
problem. 

Construction.  As  the  elements  of  the  surfaces  depicted  in 
Fig.  206  are  parallel  to  the  plane  of  the  paper,  they  are  therefore, 
shown  in  their  true  length.  Hence,  to  develop  the  surfaces, 


D' 


FIG.  206. 


FIG.  207. 


rectify  the  bases  and  set  off  the  points  at  proper  distances  from 
the  base  line  and  also  their  rectified  distance  apart.  The 
appearance  of  the  completed  developments  are  shown  in  Fig.  207. 
This  problem  is  similar  to  the  development  of  the  steam 
dome  on  a  locomotive  boiler. 

1206.  Problem  5.  To  find  the  line  of 
intersection  of  two  cylindrical  surfaces  of 
revolution  whose  axes  intersect  at  any  angle. 

Construction.  Fig.  208  shows  the  two 
cylindrical  surfaces  chosen.  If  a  series  of 
auxiliary  planes  be  passed  parallel  to  the 
plane  of  the  intersecting  axes  they  will  cut 
the  cylindrical  surfaces  in  rectilinear  ele- 
ments. The  elements  of  one  surface  will 
intersect  the  elements  of  the  other  surface 
in  the  required  points  of  the  line  of  inter- 
section. The  construction  shown  in  the 
figure  should  be  clear  from  the  similar  lettering  for  the  similar 
points  on  the  line  of  intersection. 


FIG.  208. 


214 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


It  may  be  desirable  to  note  that  this  method  of  locating 
the  required  points  on  the  curve  is  also  applicable  to  the  con- 
struction of  Fig.  206,  and  vice  versa. 

If  the  plane  of  the  intersecting  axes  is  not  parallel  to  the  plane 
of  the  paper,  the  construction  can  be  simplified  by  revolving  the 
cylindrical  surfaces  until  they  become  parallel.  The  method 
of  procedure  is  then  the  same  as  that  given  here.  If  the  axes 
cannot  be  made  to  lie  in  the  same  plane,  then  the  construction 
is  more  difficult.  The  method  in  such  cases  is  to  pass  a  plane 
through  one  cylinder  so  as  to  cut  in  in  elements;  and  the  same 
plane  will  cut  the  other  cylinder  in  some  line  of  intersection. 

The  intersection  or  these  will 
yield  points  on  the  desired  line 
of  intersection. 


c" 


FIG.  209. 


1207.  Problem  6.  To  find 
the  developments  in  the  pre- 
ceding problem. 

Construction.  Fig.  209 
shows  the  development  as  it 
appears.  If  the  plane  of  the 
base  is  perpendicular  to  the 
elements  of  the  surface,  the 
base  will  develop  into  a  straight 
line.  The  elements  will  be  at 
right  angles  to  the  base  line 
and  as  they  are  shown  in 


their  true  length,  they  may  be  laid  off  directly  from  Fig.  208. 


FIG.  210. 


FIG.  211. 


1208.  Application  of  intersecting  cylindrical  surfaces  to 
pipes.  The  construction  of  pipe  fittings  as  commercially  used 
furnish  examples  of  intersecting  cylindrical  surfaces.  Fig.  210 


INTERSECTIONS  OF   SURFACES  WITH  EACH  OTHER    215 


shows  two  cylindrical  surfaces  whose  diameters  are  the  same  and 
whose  axes  intersect  at  a  right  angle.  The  line  of  intersection 
for  this  case  will  be  seen  to  consist  of  two  straight  lines  at  right 
angles  to  each  other.  In  Fig.  211,  the  cylindrical  surfaces  have 
their  axes  intersecting  at  an  angle  of  45°  and  have  different 
diameters. 

1209.  Problem  7.  To  find  the  line  of  intersection  of  two 
cylindrical  surfaces  whose  axes  do  not  intersect. 

Construction.  Fig.  212  shows  two  circular  cylindrical  sur- 
faces which  are  perpendicular  to  each  other  and  whose  axes  are 
offset.  Pass  a  series  of  planes  ab,  cd,  ef,  etc.,  through  o.  These 
planes  cut  the  cylindrical  surface  in  elements,  which  in  turn, 
intersect  elements  of  the  other  surface.  As  the  construction 
lines  are  completely  shown,  the  description  is  unnecessary. 


FIG.  212. 


FIG.  213. 


1210.  Problem  8.  To  find  the  developments  in  the  pre- 
ceding problenio 

Construction.  First  consider  the  larger  cylindrical  surface, 
D',  Fig.  213,  which  is  cut  along  any  element.  With  a  divider, 
space  off  the  rectified  distances  between  elements;  and  on  the 
proper  elements,  lay  off  b'"  to  correspond  with  b" ;  k"'d'"  =k"d" 
i'"f'"=i"f",  etc. 

For  the  smaller  cylindrical  surface,  rectify  the  entire  circle 
ace  ...  la  and  divide  into  the  number  of  parts  shown.  .  As  the 
elements  are  shown  parallel  to  the  plane  of  projection,  in  the 
side  elevation,  they  may  be  directly  plotted  as  indicated  by  the 
curve  g'"i'"k'".  .  .  e'"g"'.  This  completes  the  final  development. 


216  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

1211.  Intersection  of  conicaFsurfaces.  If  a  plane  be  passed 
through  the  vertex  of  a  conical  surface  it  cuts  it  in  rectilinear 
elements  if  at  all.  When  two  conical  surfaces  intersect  it  is 
possible  to  pass  a  plane  through  the  vertices  of  both.  Thus, 
the  plane  may  be  made  to  intersect  both  surfaces  in  rectilinear 
elements,  the  intersections  of  which  determine  points  on  the 
required  line  of  intersection. 

Fig.  214  shows  this  pictorially.  If  the  vertices  m  and  n  be 
joined  by  a  line  and  its  piercing  point  o  be  found,  then,  any  line 
through  o  will  determine  a  plane.  Also,  if  od  be  the  line  chosen, 
the  plane  of  the  lines  mo  and  od  will  cut  the  conical  surfaces  in 


FIG.  214. 

the  elements  ma,  mb,  nc  and  nd.  These  elements  determine 
the  four  points  e,  f,  g,  h.  Any  other  line  through  o  will,  if  prop- 
erly chosen,  determine  other  elements  which  again  yield  new 
points  on  the  line  of  intersection.  The  following  problem  will 
bring  out  the  details  more  fully. 

1212.  Problem  9.  To  find  the  line  of  intersection  of  the 
surfaces  of  two  cones,  whose  bases  may  be  made  to  lie  in  the 
same  plane,  and  whose  altitudes  differ. 

Construction.  Let  A  and  B,  Fig.  215  be  the  vertices  of  the 
two  cones  in  question,  whose  bases  are  in  the  horizontal  plane. 
If  the  bases  do  not  lie  in  the  principal  planes,  a  new  set  of  prin- 
cipal planes  may  be  substituted  to  attain  the  result.  Join  A 
and  B  by  a  line  and  find  where  this  line  pierces  the  horizontal 


INTERSECTIONS  OF  SURFACES  WITH  EACH  OTHER    217 

plane  at  c.  Any  line,  through  c,  lying  in  the  horizontal  plane, 
in  addition  to  the  line  AB  will  determine  a  plane,  which  may 
cut  the  conical  surfaces  in  elements.  Let,  for  instance,  cd  be 
such  a  line.  This  line  cuts  the  bases  at  d,  e  and  f,  the  points 
which  will  be  considered.  The  elements  in  the  horizontal  pro- 
jection are  shown  as  db,  eb  and  fa;  in  the  vertical  projection, 
they  are  d'b',  e'b'  and  fa'.  The  element  fa'  intersects  the  element 
d'b'  at  g',  and  the  element  e'b'  at  h',  thus  determining  g'  and 
h',  two  points  on  the  vertical  projection  of  the  required  line  of 
intersection.  The  corresponding  projections  g  and  h  may  also 
be  found,  which  determine  points  on  the  horizontal  projection 


FIG.  215. 


of  the  same  line  of  intersection.  Consider,  now,  the  line  ci, 
tangent  to  the  one  cone.  Its  element  is  ib  in  the  horizontal 
projection  and  i'b'  in  the  vertical  projection;  the  corresponding 
element  cut  from  the  other  cone  is  ja  in  the  horizontal  projec- 
tion and  j'a'  in  the  vertical  projection.  The  points  k  and  k' 
are  thus  found,  but  they  represent  only  one  point  K  on  the  actual 
cones. 

This  process  is  continued  until  a  sufficient  number  of  points 
are  determined,  so  that  a  smooth  curve  can  be  drawn  through 
them.  The  cones  chosen,  have  complete  interpenetration,  as 
one  cone  goes  entirely  through  the  other.  Thus,  there  are  two 
distinct  curves  of  intersection.  That,  near  the  apex  B,  is  iound 
in  an  identical  way  with  the  preceding. 


218 


GEOMETRICAL  PROBLEMS  IN   PROJECTION 


Attention  might  profitably  be  called  to  the  manner  in  which 
points  are  located  so  as  to  miminize  confusion  as  much  as  possible. 
To  do  this,  draw  only  one  element  on  each  cone  at  a  time,  locat- 
ing one,  or  two  points,  as  the  case  may  be;  then  erase  the  con- 
struction* lines  when  satisfied  of  the  accuracy.  Several  of  the 
prominent  points  of  the  curve  may  be  thus  located  and  others 
estimated  if  considerable  accuracy  is  not  a  prerequisite.  If 
the  surfaces  are  to  be  developed  and  constructed  subsequently, 
however,  more  points  will  have  to  be  established. 

There  is  nothing  new  in  the  development  of  these  cones,  the 
case  is  similar  to  that  of  any  oblique  cone  and  is  therefore  omitted. 
One  thing  may  be  mentioned  in  passing  however,  and  that  is, 
while  drawing  the  elements  to  determine  the  contour  of  the  base, 
the  same  elements  should  be  used  for  locating  the  line  of  inter- 
section as  thereby  considerable  time  is  saved. 

1213.  Problem  10.  To  find  the  line  of  intersection  of  the 
surfaces  of  two  cones,  whose  bases  may  be  made  to  lie  in  the 
same  plane  and  whose  altitudes  are  equal. 

Construction.     Let  A  and  B,  Fig.  216,  be  the  desired  cones. 

Join  A  and  B  by  a  line  which  is 
chosen,  parallel  to  the  planes  of 
projection.  Hence,  this  line  cannot 
pierce  the  plane  of  the  bases,  and 
the  preceding  method  of  finding 
the  line  of  intersection  is  thus 
inapplicable.  It  is  possible,  how- 
ever to  draw  a  line  cd  parallel  to 
ab.  These  lines  therefore  determine 
a  plane  which  passes  through  both 
vertices  and  intersects  the  surface  in 
rectilinear  elements.  The  comple- 
tion of  the  construction  becomes 
evident  when  the  process  which 
leads  to  the  finding  of  F  and  G  is  understood. 

The  cases  of  intersecting  cones  so  far  considered  have  been 
so  situated  as  to  have  bases  in  a  common  plane.  This  may 
not  always  be  convenient.  Therefore  to  complete  the  subject, 
and  to  cover  emergencies,  an  additional  construction  will  be 
studied. 


FIG.  216. 


INTERSECTIONS  OF  SURFACES  WITH  EACH  OTHER    219 

1214.  Problem  11.  To  find  the  line  of  intersection  of  the 
surfaces  of  two  cones  whose  bases  lie  in  different  planes. 

Construction.  Let  A  and  B,  Fig.  217,  be  the  vertices  of  the 
two  cones.  The  cone  A  has  its  base  in  the  horizontal  plane;  B 
has  its  base  in  the  plane  T,  which  is  perpendicular  to  the  vertical 
plane.  Join  the  vertices  A  and  B  by  a  line,  which  pierces  the 
horizontal  plane  at  c  and  the  plane  T  in  the  point  d'  horizontally 
projected  at  d.  Revolve  the  plane  T  into  the  vertical  plane, 
and,  hence,  the  trace  Tt  becomes  Tt".  The  angle  t'Tt"  must 


FIG.  217. 

be  a  right  angle,  because  it  is  perpendicular  to  the  vertical  plane, 
and,  therefore  cuts  a  right  angle  from  the  principal  planes.  The 
piercing  point  of  the  line  AB  is  at  d"  in  the  revolved  position, 
where  d'd"  is  equal  to  the  distance  of  d  from  the  ground  line. 
In  the  revolved  position  of  the  plane  T,  draw  the  base  of  the 
cone  B  and  from  it  accurately  construct  the  horizontal  projec- 
tion of  the  base.  Draw  a  line  d"e"  tangent  to  the  revolved 
position  of  the  base  of  the  cone  B  at  t" .  From  it,  find  f,  the 
horizontal,  and  f,  the  vertical  projection  of  this  point,  and  draw 
the  elements  fb  and  f'b'.  Lay  off  Te"=Te,  and  draw  ce.  The 
lines  CD  and  DE  determine  a  plane  which  cuts  the  horizontal 


220 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


projection  of  the  base  of  the  cone  A  in  g  and  h,  and  thus,  also, 
the  two  elements  ag  and  ah  which  are  horizontal  projections, 
and  a'g'  and  a'h'  the  corresponding  vertical  projections  of  the 
elements.  The  element  BF  intersects  the  elements  AG  and  AH 
in  the  points  M  and  N  shown,  as  usual,  by  their  projections. 
M  and  N  are  therefore  two  points  of  the  required  curve,  being 
prominent  points  because  the  elements  are  tangent  at  these  points. 
To  obtain  other  points,  draw  a  line  d"i"  and  make  Ti"  =  Ti.  Where 
ci  cuts  the  cone  A,  draw  the  elements  as  shown,  also  the  correspond- 


FIG.  217. 

ing  elements  of  the  cone  B.     The  points  of  the  curve  of  intersec- 
tion are  therefore  as  indicated. 

One  fact  is  to  be  observed  in  this  and  in  similar  constructions. 
Auxiliary  planes  are  passed  through  the  vertices  of  both  cones, 
cutting,  therefore,  elements  of  the  cones  whose  intersection 
determine  points  on  the  curve.  All  planes  through  the  vertices 
of  both  cones  must  pass  through  c  in  the  horizontal  plane  and 
through  d"  in  the  plane  T.  Also,  such  distances  as  Te"  must 
equal  Te  because  these  auxiliary  planes  cut  the  'horizontal  plane 
and  the  plane  T  on  their  line  of  intersection  Tt;  the  revolution 
of  the  plane  T  into  the  vertical  plane  does  not  disturb  the  location 


INTERSECTIONS  OF  SURFACES  WITH  EACH 'OTHER    221 

of  any  point  on  it.     Hence  such  distance  as  Te  will  be  revolved 
to  Te"  where  Te  =  Te". 

When  applying  this  problem  to  a  practical  case,  it  would  be 
better  to  select  a  profile  plane  for  the  plane  T,  as  then  It"  would 
coincide  with  the  ground  line  and  all  construction  would  be 
simplified.  It  has  not  been  done  in  this  instance,  in  order  to 
show  the  generality  of  the  method,  and  its  adaptation  to  any 
condition. 

1215.  Types  of  lines  of  intersection  for  surfaces  of  cones. 

When  the  surfaces  of  two  cones  are  situated  so  that  there  is  com- 


FIG.  218. 


FIG.  219. 


plete  interpenetration,  the  line  of  intersection  will  appear  as 
two  distinct  closed  curves.  Fig.  215  is  an  example. 

If  the  surfaces  of  the  cones  are  such  that  the  interpenetration  is 
incomplete,  only  one  closed  curve  will  result.  Figs.  216  and  217 
are  examples  of  this  case. 

When  the  surfaces  of  two  cones  have  a  common  tangent  plane 
then  the  curve  is  closed  and  crosses  itself  once.  An  example 
under  this  heading  is  given  in  Fig.  218. 

It  is  possible  to  have  two  cones  the  surfaces  of  which  have 
two  common  tangent  planes.  In  this  case  there  are  two  closed 
curves  which  cross  each  other  twice.  This  illustration  appears 
in  Fig.  219. 


222 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


1216.  Problem  12.  To  find  the  line  of  intersection  of  the 
surfaces  of  a  cone  and  a  cylinder  of  revolution  when  their  axes 
intersect  at  a  right  angle. 

Construction.  Let  Fig.  220  represent  the  cone  and  the 
cylinder.  Through  the  cone,  pass  a  series  of  planes  perpendicular 
to  its  axis.  If  the  planes  are  properly  chosen,  they  will  cut  the 
conical  surface  in  circles  and  the  cylindrical  surface  in  rectilinear 
elements,  the  intersection  of  which  determine  points  on  the  line 
of  intersection.  A  reference  to  Fig.  220  will  show  this  in  con- 
struction. As  a  check  on  the  accuracy  of  the  points  on  the  curve, 
it  is  possible  to  draw  elements  of  the  conical  surface  through  some 
point;  the  corresponding  projections  of  the  elements  must  con- 


FIG.  220. 

ta'n  the  corresponding  projections  of  the  points.  In  the 
illustration,  the  elements  OA  and  OB  are  drawn  through  the 
points  E  and  J  respectively. 

1217.  Problem  13.  To  find  the  line  of  intersection  of  the 
surfaces  of  a  cone  and  a  cylinder  of  revolution  when  their  axes 
intersect  at  any  angle. 

Construction.  The  given  data  is  shown  in  Fig.  221.  In 
this  casexit  is  inconvenient  to  pass  planes  perpendicular  to  the 
axis  of  the  cone,  since  ellipses  will  be  cut  from  the  cylinder.  A 
better  plan  is  to  find  m,  the  intersection  of  their  axes,  and  use 
this  as  a  centre  for  auxiliary  spherical  surfaces.  The  spherical 
surfaces  intersect  the  surfaces  of  revolution  in  circles  (1025). 


INTERSECTIONS  OF  SURFACES  WITH  EACH  OTHER    223 

The  "details  of  the  construction  are  shown  in  Fig.  221.  To 
check  the  accuracy  of  the  construction  it  is  possible  to  draw  an 
element  of  one  surface  through  some  point  and  then  find  the 
corresponding  projection  of  the  element;  the  corresponding 
projection  of  the  point  must  be  located  on  the  corresponding 
projection  of  the  element.  Two  elements  OF  and  OG  are  shown 
in  the  figure.  The  points  determined  thereby  are  shown  at  D. 

1218.  Problem  14.  To  find  the  line  of  intersection  of  the 
surfaces  of  an  oblique  cone  and  a  right  cylinder. 

Construction.    Let  Fig.  222  illustrate  the  conditions  assumed. 


FIG.  222. 


To  avoid  too  many  construction  lines,  the  figures  have  been 
assumed  in  their  simplest  forms.  Pass  any  plane  through  o, 
perpendicular  to  the  horizontal  plane;  it  cuts  elements  DO  and 
CO  from  the  cone  and  the  element  from  the  cylinder  which  is 
horizontally  projected  at  f  and 'vertically  at  f'g7.  In  the  vertical 
projection,  the  elements  c'o'  and  d'o'  intersect  the  element  f'g' 
at  g'  and  h',  two  points  of  the  required  curve.  The  extreme 
element  OE  determines  the  point  i  by  the  same  method.  There 
are  two  distinct  lines  of  intersection,  in  this  case,  due  to  complete 
interpenetration.  The  points  on  the  line  of  intersection  near  the 
vertex  are  located  in  a  manner  similar  to  that  shown. 


224 


GEOMETEICAL  PROBLEMS  IN  PROJECTION 


1219.  Problem  15.  To  find  the  developments  in  the  pre- 
ceding problem. 

Construction.  The  oblique  conical  surface  is  developed 
by  triangulation.  In  Figs.  222  and  223,  the  lines  o'a'  and  o'b' 
are  shown  in  their  true  length  in  the  vertical  projection;  hence, 
lay  off  o'a'  =  o"a",  and  o'b'  =  o"b".  If  the  element  oe  be  revolved 
so  that  it  is  parallel  to  the  vertical  plane,  the  point  i',  in  the  vertical 
projection,  will  not  change  its  distance  above  the  horizontal 
plane  during  the  revolution.  Hence  it  moves  from  i'  to  k',  the 
revolved  position.  Accordingly,  lay  off  o"i"  =  o'k'  and  one 
point  of  the  intersection  on  the  development  is  obtained.  Other 
points,  of  course,  are  found  in  absolutely  the  same  manner. 


*         D 


FIG.  223. 

Extreme  accuracy  is  required  in  most  of  these  problems.  Con- 
structions like  this  should  be  laid  out  to  as  large  a  scale  as  con- 
venient. The  development  of  the  cylinder  is  also  shown  and  is 
perhaps  clear  without  explanation. 

1220.  Problem  16.  To  find  the  line  of  intersection  of  the 
surfaces  of  an  oblique  cone  and  a  sphere. 

Construction.  Let  Fig.  224  represent  the  cone  and  sphere 
in  question.  The  general  scheme  is  to  pass  planes  through  the 
vertex  of  the  cone  perpendicular  to  the  horizontal  plane.  These 
planes  cut  the  cone  in  rectilinear  elements  and  the  sphere  in 
circles;  the  intersection  of  the  elements  and  the  circles  so  cut 
will  determine  points  on  the  curve. 

Thus,  through  o,  draw  a  plane  which  cuts  the  cone  in  a  and  b 
and  the  sphere  in  d  and  e.  The  elements  cut  are  shown  as  o'a' 


INTERSECTIONS  OF  SURFACES  WITH  EACH  OTHER  225 


and  o'b'  in  the  vertical  projection.  Attention  will  be  confined 
to  the  determination  of  g',  one  point 
on  the  lower  curve,  situated  on  the 
element  OA.  The  circle  cut  from  the 
sphere  by  the  plane  through  oa  and 
ob  has  a  diameter  equal  to  de.  If  pc 
is  a  perpendicular  to  de  from  p  the 
centre  of  the  sphere,  then  c  is  the  hori- 
zontal projection  of  the  centre  of  the 
circle  de,  and  cd  and  ce  are  equal. 
If  the  cutting  plane  is  revolved  about 
a  perpendicular  through  o  to  the  hori- 
zontal plane,  until  it  is  parallel  to  the 
vertical  plane,  a  will  move  to  a"  and 
o'a'"  will  be  the  revolved  position  of 
this  element.  The  centre  of  the  circle  _ 
will  go  to  h  in  the  horizontal  projec- 
tion and  h'  in  the  vertical  projection, 
because,  in  this  latter  case,  the  distance 
of  h'  above  the  horizontal  plane  does 
not  change  in  the  revolution.  The 

element  and  the  centre  of  the  circle  in  a  plane  parallel  to  the 
vertical  plane  are  thus  determined.  Hence,  with  cd  as  a  radius 
and  h'  as  a  centre,  describe  an  arc,  cutting  o'a'"  in  f .  On  counter 

revolution,  f  goes  to  g'  on  o'a',  the 
original  position  of  the  element.  The 
point  g'  is  therefore  one  point  on 
the  curve.  Every  other  point  is 
found  in  the  same  way. 

1221.  Problem  17.  To  find  the 
line  of  intersection  of  the  surfaces 
of  a  cylinder  and  a  sphere. 

Construction.  Fig.  225  pic- 
tures the  condition.  Pass  a  series  of 
planes,  through  the  cylinder  and  the 
sphere,  perpendicular  to  the  plane  of  the  base  of  the  cylinder.  One 
position  of  the  cutting  plane  cuts  the  cylinder  in  a'b'  and  the 
sphere  in  c'd'.  Construct  a  supplementary  view  S  with  the  centre 
of  the  sphere  at  o"  as  shown.  The  elements  appear  as  a"a"  and 
b"b".  The  diameter  of  the  circle  cut  from  the  sphere  is  c'd'  and 


FIG.  224. 


FIG.  225. 


226 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


with  c'e'  as  a  radius,  (equal  to  one-half  of  c'd'),  draw  an  arc  cutting 
the  elements  at  f"g''h"  and  i".  Lay  off  f"j"  =  fj;  g"j"  =  gj,  etc., 
and  the  four  points  f,  g,  h,  i  are  determined  on  the  required 
view.  These  points  are  on  the  required  line  of  intersection. 

1222.  Problem  18.  To  find  the  line  of  intersection  of  two 
doubly  curved  surfaces  of  revolution  whose  axes  intersect. 

Construction.  Let  Fig.  226  represent  the  surfaces  in  ques- 
tion. With  o,  the  intersection  of  the  axes,  as  a  centre,  draw 
a  series  of  auxiliary  spherical  surfaces.  One  of  these  spherical 
surfaces  cuts  the  surface  whose  axis  is  on  in  a  circle  whose  diameter 
is  cd.  This  same  auxiliary  sphere  cuts  the  surface  whose  axis  is 


FIG.  226. 

om  in  a  circle  projected  as  ef.  Hence  cd  and  ef  intersect  at  a, 
a  point  on  the  required  line  of  intersection. 

In  the  view  on  the  right,  only  one  of  the  surfaces  is  shown. 
The  location  of  the  corresponding  projections  of  the  line  of 
intersection  will  be  evident  from  Art.  1027. 

When  the  axes  do  not  intersect  then  the  general  method  is 
to  pass  planes  so  as  to  cut  circles  from  one  surface  of  revolution 
and  a  curve  from  the  other.  The  intersections  determine  points 
on  the  curve.  It  is  desirable  in  this  connection  to  chose  an  arrange- 
ment that  gives  the  least  trouble.  No  general  method  can  be 
given  for  the  mode  of  procedure. 

1223.  Commercial  application  of  methods.  In  practice, 
it  is  always  desirable  as  a  matter  of  time  to  turn  the  objects  so 
that  the  auxiliary  surfaces  may  be  passed  through  them  with 
the  least  effort.  Frequently,  many  constructions  may  be  car- 
ried out  without  any  special  reference  to  the  principal  planes. 


INTERSECTIONS  OF  SURFACES  WITH  EACH  OTHER  227 

There  is  no  harm  in  omitting  the  principal  planes,  but  the  student 
should  not  go  to  the  extreme  in  this  ommission.  With  the 
principal  planes  at  hand,  the  operations  assume  a  familiar  form, 
which  will  have  a  tendency  to  refresh  the  memory  as  to  the  basic 
principles  involved.  All  the  operations  in  the  entire  subject 
have  a  remarkable  simplicity  in  the  abstract;  the  confusion 
that  sometimes  arises  is  not  due  to  the  principles  involved,  but 
solely  to  the  number  of  construction  lines  required.  It,  therefore, 
seems  proper  to  use  such  methods  as  will  lead  to  the  least  con- 
fusion, but  it  should  always  be  borne  in  mind  that  accuracy  is 
important  at  all  hazards. 

QUESTIONS  ON  CHAPTER  XII 

1.  State  the  general  method  of  finding  the  intersection  of  any  two 

surfaces. 

2.  When  the  surfaces  are  those  of  prisms  or  pyramids,  is  it  necessary 

to  use  auxiliary  planes  as  cutting  planes?    Why? 

3.  Prove  the  general  case  of  finding  the  line  of  intersection  of  the  sur- 

faces of  two  prisms. 

4.  Show  the  general  method  of  finding  the  development  of  the  prisms 

in  Question  3. 

5.  Prove  the  general  case  of  finding  the  line  of  intersection  of  two 

cylindrical  surfaces  of  revolution  whose  axes  intersect  at  a  right 
angle. 

6.  Show  the  general  method  of  finding  the  development  of  the  surfaces 

in  Question  5. 

7.  Prove  the  general  case  of  finding  the  line  of  intersection  of  two 

cylindrical  surfaces  of  revolution  whose   axes  intersect   at   any 
angle. 

8.  Show  the  general  method  of  finding  the  development  of  the  surfaces 

in  Question  7. 

9.  Prove  the  general  case  of  finding  the  line  of  intersection  of  two 

cylindrical  surfaces  whose  axes  do  not  intersect. 

10.  Show  the  general  method  of  finding  the  development  of  the  cylin- 

drical surfaces  in  Question  9. 

11.  State  the  general  method  of  finding  the  line  of  intersection  of  two 

conical  surfaces. 

12.  Prove  the  general  case  of  finding  the  line  of  intersection  of  the  sur- 

faces of  two  oblique  cones  whose  bases  may  be  made  to  be  in 
the  same  plane  and  whose  altitudes  differ. 

13.  Show  the  general  method  of  finding  the  developments  of  the  surfaces 

of  the  cones  in  Question  12. 

14.  Prove  the  general  case  of  finding  the  line  of  intersection  of  the  sur- 

faces of  two  oblique  cones  whose  bases  may  be  made  to  lie  in  the 
same  plane  and  whose  altitudes  are  equal. 


228  GEOMETRICAL  PROBLEMS  IN  PROJECTION 

15.  Show   the  general  method   of  finding    the  developments   of  the 

surfaces  of  the  cones  in  Question  14. 

16.  Prove  the  general  case  of  finding  the  line  of  intersection  of  the  sur- 

faces of  two  oblique  cones  whose  bases  lie  in  different  planes. 

17.  Show  the  general  method  of  finding  the  developments  of  the  cones 

in  Question  16. 

18.  When  the  surfaces  of  two  cones  have  complete  interpenetration, 

discuss  the  nature  of  the  line  of  intersection. 

19.  When  the  surfaces  of  two  cones  have  incomplete  penetration,  discuss 

the  nature  of  the  line  of  intersection. 

20.  When  the  surfaces  of  two  cones  have  a  common  tangent  plane,  discuss 

the  nature  of  the  line  of  intersection. 

21.  When  the  surfaces  of  two  cones  have  two  common  tangent  planes, 

discuss  the  nature  of  the  line  of  intersection. 

22.  Prove  the  general  case  of  finding  the  line  of  intersection  of  the  sur- 

faces of  a  cone  and  a  cylinder  of  revolution  when  their  axes  inter- 
sect at  a  right  angle. 

23.  Show  the  general  method  of  finding  the  developments  of  the  surfaces 

in  Question  22. 

24.  Prove  the  general  case  of  finding  the  line  of  intersection  of  the  sur- 

faces of  a  cone  and  a  cylinder  of  revolution  when  their  axes  inter- 
sect at  any  angle. 

25.  Show  the  general  method  of  finding  the  developments  of  the  sur- 

faces in  Question  24. 

26.  Prove  the  general  case  of  finding  the  line  of  intersection  of  the  sur- 

faces of  an  oblique  cone  and  a  right  cylinder. 

27.  Show  the  general  method  of  finding  the  developments  in  Question  26. 

28.  Prove  the  general  case  of  finding  the  line  of  intersection  of  the  sur- 

faces of  an  oblique  cone  and  a  sphere. 

29.  Show  the  general  method  of  finding  the  development  of  the  cone  in 

Question  28. 

30.  Develop  the  surface  of  the  sphere  in   Question  28  by  the    gore 

method. 

31.  Prove  the  general  case  of  finding  the  line  of  intersection  of  the  sur- 

faces of  a  sphere  and  a  cylinder. 

32.  Show  the  general  method  of  finding  the  development  of  the  cylinder 

in  Question  31. 

33.  Develop  the  surface  of  the  sphere  in  Question  31  by  the  zone  method. 

34.  Prove  the  general  case  of  finding  the  line  of  intersection  of  two 

doubly  curved  surfaces  of  revolution  whose  axes  intersect. 

35.  Develop  the  surfaces  of  Question  34  by  the  gore  method. 

36.  Develop  the  surfaces  of  Question  34  by  the  zone  method. 

37.  State  the  general  method  of  finding  the  line  of  intersection  of  two 

doubly  curved  surfaces  of  revolution  whose  axes  do  not  intersect. 

38.  What  items  are  to  be  considered  when  applying  the  principles  of 

intersections  and  developments  to  commercial  problems? 

39.  Is  it  always  necessary  to  use  the  principal  planes  when  solving  prob- 

lems relating  to  intersections  and  developments? 


INTERSECTIONS  OF  SURFACES  WITH  EACH  OTHER    229 

40.  Find  the  intersection  of  the  two  prisms  shown  in  Fig.  12-A.    Assume 

suitable  dimensions. 

41.  Develop  the  prisms  of  Question  40. 

42.  Two  cylinders  of  2"  diameter  intersect  at  a  right  angle.     Find  the 

line  of  intersection  of  the  surfaces.  Assume 
suitable  dimensions  for  their  lengths  and  position 
with  respect  to  each  other. 

43.  Develop  the  surfaces  of  Question  42. 

44.  Two  cylinders   of   2"  diameter  and   If"  diameter 

intersect  at  a  right  angle.  Find  the  line  of 
intersection  of  the  surfaces.  Assume  suitable 
dimensions  for  their  lengths  and  position  with 
respect  to  each  other. 

45.  Develop  the  surfaces  of  Question  44. 

46.  Two  cylinders  of  2"  diameter  intersect  at  an  angle        FlG   12_A 

of  60°.     Find  the  line  of  intersection  of  the  sur- 
faces.    Assume  suitable  dimensions  for  their  lengths  and  position 
with  respect  to  each  other. 

47.  Develop  the  surfaces  of  Question  46. 

48.  Two  cylinders  of  2"  diameter  intersect  at  an  angle  of  45°.     Find 

the  line  of  intersection  of  the  surfaces.  Assume  suitable  dimen- 
sions for  their  lengths  and  position  with  respect  to  each  other. 

49.  Develop  the  surfaces  of  Question  48. 

50.  Two  cylinders  of  2"  diameter  intersect  at  an  angle  of  30°.     Find 

the  line  of  intersection  of  the  surfaces.  Assume  suitable  dimen- 
sions for  their  lengths  and  position  with  respect  to  each  other. 

51.  Develop  the  surfaces  of  Question  50. 

52.  Two  cylinders  of  2"  diameter  and  If"  diameter  intersect  at  an 

angle  of  60°.  Find  the  line  of  intersection  of  the  surfaces.  As- 
sume suitable  dimensions  for  their  lengths  and  position  with 
respect  to  each  other. 

53.  Develop  the  surfaces  of  Question  52. 

5?-.  Two  cylinders  of  2"  diameter  and  If"  diameter  intersect  at  an  angle  of 
45°.  Find  the  line  of  intersection  of  the  surfaces.  Assume  suitable 
dimensions  for  their  lengths  and  position  with  respect  to  each  other. 

55.  Develop  the  surfaces  of  Question  54. 

56.  Two  cylinders  of  2"  diameter  and  If"  diameter  intersect  at  an  angle 

of  30°.  Find  the  line  of  intersection  of  the  surfaces.  Assume 
suitable  dimensions  for  their  lengths  and  position  with  respect 
to  each  other. 

57.  Develop  the  surfaces  of  Question  56. 

58.  Two  cylinders  of  2"  diameter  intersect  at  a  right  angle  and  have 

their  axes  offset  \" .  Find  the  line  of  intersection  of  the  surfaces. 
Assume  the  additional  dimensions. 

59.  Develop  the  surface  of  one  of  the  cylinders  of  Question  58. 

60.  A  2"  cylinder  intersects  a  If"  cylinder  so  that  their  axes  are  offset 

|"  and  make  a  right  angle  with  each  other.  Find  the  line  of 
intersection  of  the  surfaces.  Assume  the  additional  dimensions. 


230  GEOMETRICAL  PROBLEMS  IN   PROJECTION 

61.  Develop  the  surfaces  of  Question  60. 

62.  Two  2"  cylinders  have  their  axes  offset  \" .    The  elements  intersect 

at  an  angle  of  60°.     Find  the  line  of  intersection  of  the  surfaces. 
Assume  the  additional  dimensions. 

63.  Develop  the  surfaces  of  Question  62. 

64.  Two  2"  cylinders  have  their  axes  offset  J".    The  elements  intersect 

at  an  angle  of  45°.     Find  the  line  of  intersection  of  the  surfaces. 
Assume  the  additional  dimensions. 

65.  Develop  the  surfaces  of  Question  64. 

66.  Two  2"  cylinders  have  their  axes  offset  \" .    The  elements  intersect 

at  an  angle  of  30°.     Find  the  line  of  intersection  of  the  surfaces. 
Assume  the  additional  dimensions. 

67.  Develop  the  surfaces  of  Question  66. 

68.  A  2"  cylinder  intersects  a  If"  cylinder  at  an  angle  of  60°.    Their 

axes  are  offset  \" .     Find  the  line  of  intersection  of  the  surfaces. 
Assume  the  additional  dimensions. 
99.  Develop  the  surfaces  of  Question  68. 


FIG.  12-B.  FIG.  12-C. 

70.  A  2"  cylinder  intersects  a  If"  cylinder  at  an  angle  of  45°.    Their 

axes  are  offset  \".    Find  the  line  of  intersection  of.  the  surfaces. 
Assume  the  additional  dimensions. 

71.  Develop  the  surfaces  of  Question  70. 

72.  A  2"  cylinder  intersects  a  If"  cylinder  at  an  angle  of  30°.    Their 

axes  are  offset  £".    Find  the  line  of  intersection  of  the  surfaces. 
Assume  the  additional  dimensions. 

73.  Develop  the  surfaces  of  Question  72. 

74.  Find  the  line  of  intersection  of  the  surfaces  of  the  two  cones  shown 

in  Fig.  12-B.     Assume  suitable  dimensions. 

75.  Develop  the  surfaces  of  Question  74. 

76.  Assume  a  pair  of   cones  similar  to  those  shown  in  Fig.  12-B,  but, 

with  equal  altitudes.     Then,  find  the  line  of  intersection  of  the 
surfaces. 

77.  Develop  the  surfaces  of  Question  76. 

78.  Assume  a  cone  and  cylinder  similar  to  that  shown   in  Fig.  12-C. 

Then,  find  the  line  of  intersection  of  the  surfaces. 

79.  Develop  the  surfaces  of  Question  78. 


INTERSECTIONS   OF  SURFACES  WITH  EACH  OTHER    231 


80.  Assume  a  cone  and  cylinder  similar  to  that   shown  in   Fig.  12-C, 

but,  have  the  cylinder  elliptical.  Then,  find  the  line  of  inter- 
section of  the  surfaces. 

81.  Develop  the  surfaces  of  Question  80. 

82.  Assume  a  cone  and  cylinder  similar  to   that  shown  in  Fig.  12-C, 

but,  have  the  cone  elliptical.  Then,  find  the  line  of  intersection 
of  the  surfaces. 

83.  Develop  the  surfaces  of  Question  82. 

84.  Assume  an    arrangement  of    cone  and  cylinder  somewhat  similar 

to  that  of  Fig.  12-C,  but,  have  both  cone  and  cylinder  elliptical. 
Then,  find  the  line  of  intersection  of  the  surfaces. 

85.  Develop  the  surfaces  of  Question  84. 

86.  Assume  a  cylinder  and  cone  of  revolution,  whose   general  direction 

of  axes  are  at  right  angles,  but,  which  are  offset  as  shown  in 
Fig.  12-D.  Then,  find  the  line  of  intersection  of  the  sur- 
faces. 

87.  Develop  the  surfaces  in  Question  86, 


FIG.  12-D. 


FIG.  12-E. 


88.  Assume  an  elliptical  cylinder  and  a  cone  of  revolution,  whose  general 

direction  of  axes  are  at  right  angles  to  each  other,  but,  which 
are  offset  as  shown  in  Fig.  12-D.  Then,  find  the  line  of  inter- 
section of  the  surfaces. 

89.  Develop  the  surfaces  of  Question  88. 

90.  Assume  a  circular  cylinder  of  revolution  and  an  elliptical  cone,  whose 

general  direction  of  axes  are  at  right  angles  to  each  other,  but, 
which  are  offset  as  shown  in  Fig.  12-D.  Then,  find  the  line  of 
intersection  of  the  surfaces. 

91.  Develop  the  surfaces  of  Question  90. 

92.  Assume  an  elliptical  cylinder  and  an  elliptical  cone,  whose  general 

direction  of  axes,  are  at  right  angles  to  each  other,  but,  which 
are  offset,  as  shown  in  Fig.  12-D.  Then,  find  the  line  of  inter- 
section of  the  surfaces. 

93.  Develop  the  surfaces  of  Question  92. 

94.  Assume  a  cylinder  and  cone  of  revolution,  similar  to  that  of   Fig. 

12-E,  and  make  angle  a  =60°.  Then,  find  the  line  of  intersection 
of  the  surfaces. 

95.  Develop  the  surfaces  of  Question  94. 


232 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


96.  Assume  a  clyinder  and  cone  of  revolution,  similar  to  that  of  Fig. 

12-E,   and  make  angle  a  =45°.     Then,  find  the  line  of  inter- 
section of  the  surfaces. 

97.  Develop  the  surfaces  of  Question  96. 

89.  Assume  a  cylinder  and  a  cone  of  revolution,  similar  to  that  of  Fig. 

12-E,  and  make  angle  «  =30°.     Then,  find  the  line  of  intersection 

of  the  surfaces. 
99.  Develop  the  surfaces  of  Question  98. 

100.  Assume  an  elliptical  cylinder  and  a  cone  of  revolution  arranged 

similar  to  that  of  Fig.  12-E,  and  make  the  angle  a  =60°.     Then, 
find  the  line  of  intersection  of  the  surfaces. 

101.  Develop  the  surfaces  of  Question  100. 

102.  Assume  a  cylinder  of  revolution  and  an  elliptical  cone,  arranged 

similar  to  that  of  Fig.  12-E,  and  make  the  angle  a  =45°.     Then, 
find  the  line  of  intersection  of  the  surfaces. 

103.  Develop  the  surfaces  of  Question  102. 


FIG.  12-E. 


FIG.  12-F. 


FIG.  12-G. 


104.  Assume  an  elliptical  cylinder  and  an  elliptical  cone  arranged  similar 

to  that  of  Fig.  12-E,  and  make  the  angle  a  =60°.     Then  find 
the  line  of  intersection  of  the  surfaces. 

105.  Develop  the  surfaces  of  Question  104. 

106.  Assume  a  cylinder  and  cone  of  revolution  as  shown  in  Fig.  12-F, 

and  make  angle  a  =60°.     Then,  find  the  line  of  intersection  of 
the  surfaces. 

107.  Develop  the  surfaces  of  Question  106. 

108.  Assume  a  cylinder  and  a  cone  of  revolution  as  shown  in  Fig.  12-F, 

and  make  angle  a  =45°.     Then,  find  the  line  of  intersection  of 
the  surfaces. 

109.  Develop  the  surfaces  of  Question  108. 

110.  Assume  a  cylinder  and  a  cone  of  revolution  as  shown  in  Fig.  12-F, 

and  make  angle  a  =  30°.    Then,  find  the  line  of  intersection  of 
the  surfaces. 

111.  Develop  the  surfaces  of  Question  110. 

112.  Assume  an  elliptical  cylinder  and  a  cone  of  revolution,  arranged 

similar  to  that  shown  in  Fig.  12-F,  and  make  angle    a  =60°. 
Then  find  the  line  of  intersection  of  the  surfaces. 


INTERSECTIONS  OF  SURFACES  WITH  EACH  OTHER    233 

113.  Develop  the  surfaces  of  Question  112. 

114.  Assume  a  cylinder  of  revolution  and  an  elliptical  cone,  arranged 

similar  to  that  shown  in  Fig.   12-F,   and  make  angle  a  =45°. 
Then,  find  the  line  of  intersection  of  the  surfaces. 


FIG.  12-H. 


FIG.  12-1. 


115.  Develop  the  surfaces  of  Question  114. 

116.  Assume  an  elliptical  cylinder  and  an  elliptical  cone,   somewhat 

similar  to  that  shown  in  Fig.   12-F,  and  make  angle  «=60°. 
Then  find  the  line  of  intersection  of  the  surfaces. 

117.  Develop  the  surfaces  of  Question  116. 

118.  Assume  a  cone  and  cylinder  as  shown  in  Fig.  12-G.    Then,  find 

the  line  of  intersection  of  the  surfaces. 

119.  Develop  the  surfaces  of  Question  118. 

120.  Assume  a  cylinder  and  a  sphere  as  shown  in  Fig.   12-H.     Then, 

find  the  line  of  intersection  of   the  sur- 
faces. 

121.  Develop  the  surface  of  the  cylinder  of  Ques- 

tion 120. 

122.  Develop  the  surface  of  the  cylinder  of  Ques- 

tion 120,   and,   also,   the  surface  of  the 
sphere  by  the  gore  method. 

123.  Assume  a  cylinder  and   sphere  as   shown  in 

Fig.  12-1.     Then,  find  the  line  of  intersec- 
tion of  the  surfaces. 

124.  Develop  the  surface  of  the  cylinder  in  Ques- 

tion 123. 

125.  Develop  the  surface  of  the  cylinder  in  Ques- 

tion 123,  and,  also,  the  surface  of  the  sphere  by  the  zone  method, 

126.  Assume  a  cone  and  a  sphere  as  shown  in  Fig.  12-J.     Then,  find 

the  line  of  intersection  of  the  surfaces. 

127.  Develop  the  surface  of  the  cone  in  Question  126. 

128.  Develop  the  surface  of  the  cone  in  Question  126,  and,  also,  the 

sphere  by  the  gore  method. 


FIG.  12-J. 


234 


GEOMETRICAL  PROBLEMS  IN  PROJECTION 


129.  Assume  two    doubly  curved    surfaces  of  revolution,  whose  axes 

intersect.    Then,  find  the  line  of  intersection  of  the  surfaces. 

130.  Develop  the  surfaces  of  Question  129  by  the  gore  method. 

131.  Develop  the  surfaces  of  Question  129  by  the  zone  method. 

132.  Assume  two  doubly  curved  surfaces  of  revolution  whose  axes  do 

not  intersect.    Then,  find  the  line  of  intersection  of  the  surfaces. 


FIG.  12-K. 


133.  Develop  the  surfaces  of  Question  132  by  the  gore  method. 

134.  Develop  the  surfaces  of  Question  132  by  the  zone  method. 

135.  Assume  a  frustum  of  a  cone  as  shown  in  Fig.  12-K.    A  circular 

cylinder  is  to  be  fitted  to  it,  as  shown.     Develop  the  surfaces. 
HINT. — Find  vertex  of  cone  before   proceeding   with   the   devel- 
opment. 


PART    III 

PRINCIPLES   OF   CONVERGENT   PROJECTING-LINE 

DRAWING 


CHAPTER  XIII 
,     PERSPECTIVE     PROJECTION 

1301.  Introductory.     Observation  shows  that  the  apparent 
magnitude  of  objects  is  some  function  of  the  distance  between 
the  observer  and  the  object.     The  drawings  made  according  to 
the  principles  of  parallel  projecting-line  drawing,  and  .treated  in 
Part  I  of  this  book,  make  no  allowance  for  the  observer's  posi- 
tion with  respect  to  the  object.     In  other  words,  the  location  of 
the  object  with  respect  to  the  plane  of  projection  has  no  influence 
on  the  size  of    the   resultant  picture,  provided  that  the  inclina- 
tion of  the  various  lines  on  the  object,  to  the  plane  of  projec- 
tion, remaiivrfre  same.     Hence,  as  the  remoteness  of  the  object 
influence^' its   apparent   size,   then,   as   a   consequence,   parallel 
projecting-line   drawings   must   have   an  unnatural   appearance. 
The  strained  appearance  of  drawings  of  this  type  is  quite  notice- 
able in  orthographic  projection  wherein  two  or  more  views  must 
be  interpreted  simultaneously,  and  is  less  noticeable  in  the  case 
of  oblique  or  axonometric  projection.     On  the  other  hand,  the 
rapidity  with  which  drawings  of  the  parallel  projecting-line  type 
can  be  made,  and  their  adaptability  for  construction  purposes, 
are  strong  points  in  their  favor. 

1302.  Scenographic  projection.    To  overcome  the  foregoing 
objections,  and  to  present  a  drawing  to  the  reader  which  cor- 
rects for  distance,  scenographic  projections  are  used.    To  make 
these,  convergent  projecting  lines   are  used,  and  the    observer 
is  located  at  the  point  of  convergency.     The  surface  on  which 

235 


236  CONVERGENT  PROJECTING-LINE  DRAWING 

scenographic  projections  are  made  may  be  spherical,  cylindrical, 
etc.,  and  find  their  most  extensive  application  in  decorative 
painting. 

1303.  Linear  perspective.     In  the  decoration  of  an  interior 
of  a  dome  or  of  a  cylindrical  wall,  the  resulting  picture  should 
be  of  such  order  as  to  give  a  correct  image  for  the  assumed  loca- 
tion of  the  eye.     In  engineering  drawing,  there  is  little  or  no  use 
for  projections  on  spherical,  cylindrical,  or  other  curved  surfaces. 
When  scenographic  projections  are  made  on  a  plane,  this  type  of 
projection   is   called   linear   perspective.     A   linear   perspective, 
therefore,  may  be  defined  as  the  drawing  made  on  a  plane  surface 
by  the  aid  of  convergent  projecting  lines,  the  point  of  convergency 
being  at  a  finite  distance  from  the  object  and  from  the  plane. 
The  plane  of  projection  is  called  the  picture  plane,  and  the  posi- 
tion of  the  eye  is  referred  to  as  the  point  of  sight. 

1304.  Visual  rays  and  visual  angle.     If  an  object — an  arrow 
for  instance — be  placed  a  certain  distance  from  the  eye,  say  in 

the  position  ab  (Fig.  227)  and  c  be 
the  point  of  sight,  the  two  extreme 
rays  of  light  or  visual  rays,  ac  and 
ab,  form  an  angle  which  is  known  as 
the  visual  angle.  If,  now,  the  arrow  be 
FIG.  227.  moved  to  a  more  remote  position  de, 

the  limiting  visual  rays  dc   and   ec 

give  a  smaller  visual  angle.  The  physiological  effect  of  this 
variation  in  the  visual  angle  is  to  alter  the  apparent  size  of  the 
object.  If  one  eye  is  closed  during  this  experiment,  the  distance 
from  the  eye  to  the  object  cannot 'be  easily  estimated,  but  the 
apparent  magnitude  of  the  object  will  be  some  function  of  the 
visual  angle.  In  binocular  vision,  the  muscular  effort  required 
to  focus  both  eyes  on  the  object  will,  with  some  experience,  enable 
an  estimate  of  the  distance  from  the  eye  to  the  object;  the  mind 
will  automatically  correct  for  the  smaller  visual  angle  and  greater 
distance,  and,  thus,  to  an  experienced  observer,  give  a  more  or 
less  correct  impression  of  the  actual  size  of  the  object.  This 
experience  is  generally  limited  to  horizontal  distances  only,  as 
very  few  can  estimate  correctly  the  diameter  of  a  clock  on  a  church 
steeple  unless  they  have  been  accustomed  to  making  such  observa- 
tions. As  the  use  of  two  eyes  in  depicting  objects  in  space  causes 


PERSPECTIVE  PROJECTION  237 

slightly   different   inpressions   on    different    observers,   the    eye 
will  be  assumed  hereafter  as  a  single  point. 

1305.  Vanishing  point.     When  looking  along  a  stretch  of 
straight  railroad  tracks,  it  may  be  noted  that  the  tracks  appar- 
ently vanish  in  the  distance.     Likewise,  in  viewing  a  street  of 
houses  of  about  the  same  height,  the  roof  line  appears  to  meet 
the  sidewalk  line  in  the  distance.      It  is  known  that  the  actual 
distances  between  the  rails,  and  between  the  sidewalk  and  roof 
is  always  the  same;  yet  the  visual  angle  being  less  in  the  distant 
observation,   gives  the  impression  of  a  vanishing  point,  which 
may,  therefore,  be  defined  as  the  point  where  parallel  lines  seem 
to  vanish.     At  an  infinite  distance  the  visual  angle  is  zero,  and, 
hence,  the  parallel  lines  appear  to  meet  in  a  point.     The  lines 
themselves    do    not    vanish,  but     their    perspective    projections 
vanish. 

1306.  Theory   of  perspective  projection.     The  simplest 
notion  of  perspective  drawing  can  be  obtained  by  looking  at  a 
distant  house  through  a  window-pane.      If  the  observer  would 
trace  on  the  window-pane  exactly  what  he  sees,  and  locate  all 
points  on  the  pane  so  that  corresponding  points  on  the  house  are 
directly  behind  them,  a  true  linear  perspective  of  the  house  would 
be  the  result.     Manifestly,  the  location  of  each  successive  point 
is  the  same  as  locating  the  piercing  point  of  the  visual  ray  on 
the  picture  plane,  the  picture  plane  in  our  case  being  the  window- 
pane. 

1307.  Aerial   perspective.      If    this  perspective   were  now 
colored  to  resemble  the  house  beyond,  proper  attention  being 
paid  to  light,  shade  and  shadow,  an  aerial  perspective  would  be 
the  result.     In  brief,  this  aerial  perspective  would  present  to  the 
•eye  a  picture  that  represents  the  natural  condition  as  near  as  the 
skill  of  the  artist  will  permit.     In  our  work  the  linear  perspective 
will  alone  be  considered,  and  will  be  denoted  as  the  "  perspective," 
aerial  perspective  finding  little  application  in  engineering. 

1308.  Location  of  picture  plane.     It  is  customary  in  per- 
spective to  assume  that  the  picture  plane  is  situated  between  the 
eye  and  the  object.     Under  these  conditions,  the  picture  is  smaller 
than  the  object,  and  usually  this  is  necessary.     The  eye  is  assumed 
to  be  in  the  first  angle;   the  object,  however,  is  generally  in  the 


238 


CONVERGENT  PROJECTING-LINE  DRAWING 


second  angle  *  for  reasons  noted  above.  The  vertical  plane  is 
thus  the  picture  plane.  A  little  reflection  will  show  that  this  is 
in  accord  with  our  daily  experience.  Observers  stand  on  the 
ground  and  look  at  some  distant  object;  visual  rays  enter  the 
eye  at  various  angles;  the  mean  of  these  rays  is  horizontal  or 
nearly  so,  and,  therefore,  the  projections  naturally  fall  on  a 
vertical  plane  interposed  in  the  line  of  sight.  The  window- 
pane  picture  alluded  to  is  an  example. 

1309.  Perspective  of  a  line.  Let,  in  Fig.  228,  AB  be  an 
arrow,  standing  vertically  as  shown  in  the  second  angle;  the 
point  of  sight  C  is  located  in  the  first  angle.  The  visual  rays 
pierce  the  picture  plane  at  a"  and  b",  and  a"b",  in  the  picture 
plane,  is  the  perspective  of  AB  in  space.  The  similar  condition 


f 


FIG.  228. 


of  Fig.  228  in  orthographic  projection  is  represented  in  Fig. 
229.  The  arrow  and  the  point  of  sight  are  shown  by  their  pro- 
jections on  the  horizontal  and  vertical  planes.  The  arrow  being 
perpendicular  to  the  horizontal  plane,  both  projections  fall  at 
the  same  point  ab;  the  vertical  projection  is  shown  as  a'b';  the 
point  of  sight  is  represented  on  the  horizontal  and  vertical 
planes  as  c  and  c',  respectively.  Any  visual  ray  can  likewise  be 
represented  by  its  projections,  and,  therefore,  the  horizontal 
projection  of  the  point  of  sight  is  joined  with  the  horizontal  pro- 

*  The  object  may  be  located  in  any  angle;  the  principles  are  the  same 
for  all  angles.  It  is  perhaps  more  convenient  to  use  the  second  angle,  as 
the  lines  do  not  have  to  be  extended  to  obtain  the  piercing  points  as  would 
be  the  case  for  first-angle  projections. 


PERSPECTIVE  PROJECTION 


239 


jections.     Similarly,  the  same  is  true  of  the  vertical  projections 
of  the  arrow  and  point  of  sight. 

It  is  now  necessary  to  find  the  piercing  points  (506,805)  of 
the  visual  rays  on  the  picture  plane  (vertical  plane);  and  these 
are  seen  at  a"  and  b".  Therefore,  a"b"  is  the  perspective  of 
AB  in  space. 

1310.  Perspectives  of  lines  perpendicular  to  the  hori= 
zontal  plane.  In  Fig.  230,  several  arrows  are  shown,  in  projec- 
tion, all  of  which  are  perpendicular  to  the  horizontal  plane.  Their 
perspectives  are  a"b",  d"e",  g"h".  On  observation,  it  will 
be  noted  that  all  the  perspectives  of  lines  perpendicular  to  the 
horizontal  plane  are  vertical.  This  is  true,  since,  when  a  plane 


FIG.  230. 


FIG.  231, 


is  passed  through  any  line  and  revolved  until  it  contains  the 
point  of  sight,  the  visual  plane  (as  this  plane  is  then  called)  is 
manifestly  perpendicular  to  the  horizontal  plane,  and  its  vertical 
trace  (603)  is  perpendicular  to  the  ground  line. 

1311.  Perspectives  of  lines  parallel  to  both  principal 
planes.  Fig.  231  shows  two  arrows  parallel  to  both  planes,  the 
projections  and  perspectives  being  designated  as  before.  This 
case  shows  that  all  perspectives  are  parallel.  To  prove  this, 
pass  a  plane  through  the  line  in  space  and  the  point  of  sight; 
the  vertical  trace  of  the  plane  so  obatined  will  be  parallel  to 
the  ground  line*  because  the  lirfe  is  parallel  to  the  ground 
line. 


*  If  the  line  in  space  lies  in  the  ground  line,  the  vertical  trace  will  coincide 
with  the  ground  line.     This  is  evidently  a  special  case. 


240 


CONVERGENT  PROJECTING-LINE  DRAWING 


1312.  Perspectives  of  lines  perpendicular  to  the  picture 
plane.  Suppose  a  series  of  lines  perpendicular  to  the  vertical 
plane  is  taken.  This  case  is  illustrated  in  Fig.  232.  The  per- 
spectives are  drawn  as  before  and  designated  as  is  customary. 
It  may  now  be  observed  that  the  perspectives  of  all  these  lines 


FIG.  232. 

(or  arrows)  vanish  in  the  vertical  projection  of  the  point  of  sight. 
This  vanishing  point  for  the  perspectives  of  all  perpendiculars 
to  the  picture  plane  is  called  the  centre  of  the  picture.  The 
reason  for  this  is  as  follows:  It  is  known  that  the  visual  angle 


FIG.  233. 


FIG.  234. 


becomes  less  as  the  distance  from  the  point  of  sight  increases; 
at  an  infinite  distance  the  angle  is  zero,  the  perspectives  of  the 
lines  converge,  and,  therefore,  in  our  nomenclature  vanish. 

1313.  Perspectives    of    parallel    lines,    inclined    to    the 
picture  plane.     Assume  further  another  set  of  lines,   Fig.  233, 


PERSPECTIVE  PROJECTION  241 

parallel  to  each  other,  inclined  to  the  vertical  plane,  but  parallel 
to  the  horizontal  plane.  Their  orthographic  projections  are 
parallel  and  their  perspectives  will  be  seen  to  converge  to  the 
point  V.  Like  the  lines  in  the  preceding  paragraph,  they  vanish 
at  a  point  which  is  the  vanishing  point  of  any  number  of  lines 
parallel  only  to  those  assumed.  The  truth  of  this  is  established 
by  the  fact  that  the  visual  angle  becomes  zero  (1304)  at  an  infinite 
distance  from  the  point  of  sight  and,  hence,  the  perspectives 
vanish  as  shown.  To  determine  this  vanishing  point,  draw 
any  line  through  the  point  of  sight  parallel  to  the  system 
of  parallel  lines;  and  its  piercing  point  on  the  vertical  or 
picture  plane  will  be  the  required  vanishing  point.  It 
amounts  to  the  same  thing  to  say  that  the  vanishing  point 
is  the  perspective  of  any  line  of  this  system  at  an  infinite  dis- 
tance. 

A  slightly  different  condition  is  shown  in  Fig.  234.  The 
lines  are  still  parallel  to  each  other  and  to  the  horizontal  plane, 
but  are  inclined  to  the  vertical  plane  in  a  different  direction. 
Again,  the  perspectives  of  these  lines  will  converge  to  a  new  van- 
ishing point,  situated,  however,  much  the  same  as  the  one  just 
preceding. 

1314.  Horizon.  On  observation  of  the  cases  cited  in  Art. 
1313,  it  can  be  seen  that  the  vanishing  point  lies  on  a  horizontal 
line  through  the  vertical  projection  of  the  point  of  sight.  This 
is  so  since  a  line  through  the  point  of  sight  parallel  to  either  sys- 
tem will  pierce  the  picture  plane  in  a  point  somewhere  on  its 
vertical  projection.  As .  the  line  is  parallel  to  the  horizontal 
plane,  its  vertical  projection  will  be  parallel  to  the  ground  line 
and,  as  shown,  will  contain  all  vanishing  points  of  all  systems  of 
horizontal  lines.  Any  one  system  of  parallel  lines  will  have 
but  one  vanishing  point,  but  as  the  lines  may  slope  in  various 
directions,  and  still  be  parallel  to  the  horizontal  plane,  every 
system  will  have  its  own  vanishing  point  and  the  horizontal  line 
drawn  through  the  vertical  projection  of  the  point  of  sight  will 
be  the  locus  of  all  these  vanishing  points.  This  horizontal 
line  through  the  vertical  projection  of  the  point  of  sight  is  called 
the  horizon. 

As  a  corollary  to  the  above,  it  may  be  stated,  that  all  planes 
parallel  to  the  horizontal  plane  vanish  in  the  horizon.  As  before, 


242 


CONVERGENT  PROJECTING-LINE  DRAWING 


the  visual  angle  becomes  less  as  the  distance  increases,  and,  hence, 
becomes  zero  at  infinity. 

There  are  an  unlimited  number  of  systems  of  lines  resulting 
in  an  unlimited  number  of  vanishing  points,  whether  they  are 
on  the  horizon  or  not.  In  most  drawings  the  vertical  and  hori- 
zontal lines  are  usually  the  more  common.  The  lines  perpendicular 
to  the  picture  plane  are  horizontal  lines,  and,  therefore,  the  centre 
of  the  picture  (vanishing  point  for  the  perspectives  of  the  per- 
pendiculars) must  be  on  the  horizon. 

1315.  Perspective  of  a  point.  The  process  of  finding  the 
perspective  of  a  line  of  definite  length  is  to  find  the  perspectives 


235. 


FIG.  236. 


of  the  two  extremities  of  the  line.  When  the  perspectives  of  the 
extremities  of  the  line  are  found,  the  line  joining  them  is  the 
perspective  of  the  given  line.  In  Fig.  235,  two  points  A  and  B 
are  chosen  whose  perspectives  will  be  found  to  be  a"  and  b". 

When  the  point  lies  in  picture  plane  it  is  its  own  perspective. 
This  is  shown  in  Fig.  236,  where  a'  is  the  vertical  projection  and 
a  is  therefore  in  the  ground  line;  hence,  the  perspective  is  a'. 

In  the  construction  of  any  perspective,  the  method  is  to 
locate  the  perspectives  of  certain  points  which  are  joined  by  their 
proper  lines.  The  correct  grouping  of  lines  determine  certain 
surfaces  which,  when  closed,  determine  the  required  solid. 


1316.  Indefinite  perspective  of  a  line.     Let  AB,  Fig.  237, 
be  a  limited  portion  of  a  line  FE.     Hence,  by  obtaining  the 


PERSPECTIVE  PROJECTION 


243 


piercing  points  of  the  visual  rays  AC  and  BC,  the  perspective 
a"b"  is  determined.  Likewise,  DE  is  another  limited  portion 
of  the  line  FE  and  its  perspective  is  d"e".  If  a  line  be  drawn 
through  the  point  of  sight  C,  parallel  to  the  line  FE,  then,  its 
piercing  point  g'  on  the  picture  plane  will  be  the  vanishing  point 
of  the  perspectives  of  a  system  of  lines  parallel  to  FE  (1313). 
Also,  as  FE  pierces  the  picture  plane  at  f,  then  f  will  be 
its  own  perspective  (1315).  Therefore,  if  a  line  be  made  to 
join  f  and  g',  it  will  be  the  perspective  of  a  line  that  reaches 
from  the  picture  plane  out  to  an  infinite  distance.  And,  also, 
the  perspective  of  any  limited  portion  of  this  line  must  lie 


FIG.  237. 


on  the  perspective  of  the  line.  From  the  construction  in 
Fig.  237,  it  will  be  observed  that  a"b"  and  d"e"  lie  on  the 
line  f  g'- 

Consequently,  the  line  f'g'  is  the  indefinite  perspective  of  a 
line  FE  which  reaches  from  the  picture  plane  at  f  out  to  an 
infinite  distance.  Thus,  the  indefinite  perspective  of  a  line  may  be 
defined  as  the  perspective  of  a  line  that  reaches  from  the  picture 
plane  to  infinity. 

Before  leaving  Fig.  237,  it  is  desirable  to  note  that  g'  is  the 
vanishing  point  of  a  system  of  lines  parallel  to  FE.  It  is  not 
located  on  the  horizon  because  the  line  FE  is  not  parallel  to  the 
horizontal  plane. 


244 


CONVERGENT  PROJECTING-LINE  DRAWING 


1317.  Problem  1.  To  find  the  perspective  of  a  cube  by  means 
of  the  piercing  points  of  the  visual  rays  on  the  picture  plane. 

Construction.  Let  ABCDEFGH,  Fig.  238,  be  the  eight 
corners  of  the  cube,  and  let  S  be  the  point  of  sight.  The  cube  is 
in  the  second  angle  and  therefore  both  projections  are  above 
the  ground  line  (315,  317,  514,  516).  The  horizontal  projec- 
tion of  the  cube  is  indicated  by  abed  for  the  upper  side,  and  efgh 
for  the  base.  The  vertical  projections  are  shown  as  a'b'c'd' 
for  the  upper  side,  and  eTg'h'  for  the  base.  The  cube  is  located 
in  the  second  angle  so  that  the  two  projections  overlap.  The 
point  of  sight  is  in  the  first  angle  and  is  shown  by  its  horizontal 
projection  s  and  its  vertical  projection  s'.  Join  s  with  e  and 


FIG.  238. 

obtain  the  horizontal  projection  of  the  visual  ray  SE  in  space; 
do  likewise  for  the  vertical  projection  with  the  result  that  s'e' 
will  be  the  visual  ray  projected  on  the  picture  plane.  The  point 
where  this  pierces  the  picture  plane  is  E,  and  this  is  one  point 
of  the  required  perspective.  Consider  next  point  C.  This 
is  found  in  a  manner  similar  to  point  E  just  determined.  The 
horizontal  projection  of  the  visual  ray  SC  is  sc  and  the  vertical 
projection  of  the  visual  ray  is  s'c'  and  this  pierces  the  vertical 
or  picture  plane  in  the  point  C  as  shown.  In  the  foregoing  man- 
ner, all  other  points  are  determined.  By  joining  the  correct 
points  with  each  other,  a  linear  perspective  will  be  obtained. 

It  will  be  noticed  that  CG,  BF,  DH,  and  AE  are  vertical 
because  the  lines  in  space  are  vertical  (1310),  and  hence  their 
perspectives  are  vertical.  This  latter  fact  acts  as  a  check  after 


PERSPECTIVE  PROJECTION 


245 


locating  the  perspectives  of  the  upper  face  and  the  base  of  the 
cube.  It  will  be  noticed,  further,  that  the  perspectives  of  the 
horizontal  lines  in  space  meet  at  the  vanishing  points  V  and  V 
(1313,  1314). 

1318.  Perspectives  of  intersecting  lines.     Instead  of  locat- 
ing the  piercing  point  of  a  visual  ray  by  drawing  the  projections 
of  this  ray,  it  is  sometimes  found  desirable  to  use  another  method. 
For  this  purpose  another  principle  must  be  developed. 

If  two  lines  in  space  inter- 
sect, their  perspectives  inter- 
sect, because  the  perspective 
of  a  line  can  be  considered  as 
being  made  up  of  the  perspec- 
tives of  all  the  points  on  the 
line.  As  the  intersection  is  a 
point  common  to  the  two 
lines,  a  visual  ray  to  this  point 
should  pierce  the  picture  plane 
in  the  intersection  of  the 
perspectives.  Reference  to 
Fig.  239  will  show  that  such  j?IG  239 

is  the  case.     CG  is  the  visual 

ray  in  space  of  the  point  of  intersection  of  the  two  lines,  and  its 
perspective  is  g",  which  is  the  intersection  of  the  perspectives 
a"b"  and  d"e". 

1319.  Perpendicular  and  diagonal.  Obviously,  if  it  is  desired 
to  find  the  perspective  of  a  point  in  space,  two  lines  can  be  drawn 
through  the  piont  and  the  intersection  of  their  perspectives  found. 
The  advantage  of  this  will  appear  later.     The  two  lines  generally 
used  are:    first,  a  perpendicular,  which  is  a  line  perpendicular 
to  the  picture  plane  and  whose  perspective  therefore  vanishes 
in  the  vertical  projection  of  the  point  of  sight  (1312);  and  second, 
a  diagonal  which  is  a  horizontal  line  making  an  angle  of  45°  with 
the  picture  plane,  and  whose  perspective  vanishes  somewhere 
on    the    horizon    (1313).     As    the    perpendicular    and    diagonal 
drawn  through  a  point  are  both  parallel  to  the  horizontal  plane, 
these  two  intersecting  lines  determine  a  plane  which,  like  all 
other  horizontal  planes,  vanish  in  the  horizon.     Instead  of  using 
a  diagonal  making  45°,  any  other  angle  may  be  used,  provided 


246 


CONVERGENT  PROJECTING-LINE  DRAWING 


it  is  less  than  90°.  This  latter  line  would  be  parallel  to  the  picture 
plane  and  therefore  would  not  pierce  it. 

It  can  also  be  observed  that  there  are  two  possible  diagonals 
through  any  point,  one  whose  perspective  vanishes  to  the  left 
of  the  point  of  sight  and  the  other  whose  perspective  vanishes 
to  the  right  of  the  point  of  sight. 

It  is  further  known  that  the  perspectives  of  all  parallel  lines 
vanish  in  one  point  and  therefore  the  perspectives  of  all  parallel 
diagonals  through  any  point  must  have  a  common  vanishing 
point.  With  two  possible  diagonals  through  a  given  point  in 
space,  two  vanishing  points  are  obtained  on  the  horizon. 

1320.  To  find  the  perspective  of  a  point  by  the  method 
of  perpendiculars  and  diagonals.  Let  a  and  a',  Fig.  240, 


Horizon 


FIG.  240. 

represent  the  projections  of  a  point  in  the  second  angle,  and  let  c 
and  c'  be  the  projections  of  the  point  of  sight.  By  drawing  the 
visual  rays  ac  and  a'c'  the  piercing  point  a"  is  determined  by 
erecting  a  perpendicular  at  e  as  shown.  Although  the  perspective 
a"  is  determined  by  this  method,  it  may  also  be  determined  by 
finding  the  intersections  of  the  perspectives  of  a  perpendicular 
and  a  diagonal  through  the  point  A  in  space. 

Thus,  by  drawing  a  line  ab,  making  a  45°  angle  with  the 
ground  line,  the  horizontal  projection  of  the  diagonal  is  found. 
As  the  diagonal  is  a  horizontal  line,  its  vertical  projection  is 
a'b'.  This  diagonal  pierces  the  picture  plane  at  b',  which  point 
is  its  own  perspective.  Its  perspective  also  vanishes  at  V,  the 
vanishing  point  of  all  horizontal  lines  whose  inclination  to  the 
picture  plane  is  at  the  45°  angle  shown  and  whose  directions  are 
parallel  to  each  other.  The  point  V  is  found  by  drawing  a  line 


PERSPECTIVE  PROJECTION 


247 


cv  parallel  to  ab,  c'V  parallel  to  a'b',  and  then  finding  the  piercing 
point  V.  The  horizontal  projection  of  a  perpendicular  through 
A  is  ad;  its  vertical  projection  is  evidently  a',  which  is  also 
its  piercing  point  on  the  picture  plane  and  as  it  lies  in  the  picture 
plane,  it  is  hence  its  own  perspective.  The  vanishing  point  of  the 
perspective  of  the  perpendicular  is  at  c',  the  centre  of  the  picture. 
Since  the  perspective  of  a  given  point  must  lie  on  the  per- 
spectives of  any  two  lines  drawn  through  the  point,  then,  as 
b'V  is  the  indefinite  perspective  of  a  diagonal,  and  a'c'  is  the 
indefinite  perspective  of  a  perpendicular,  their  intersection  a"  is 
the  required  perspective  of  the  point  A.  The  fact  that  a"  has 
already  been  determined  by  drawing  the  visual  ray  and  its 


v 


FIG.  241. 

piercing  point    found  shows  that    the    construction  is  correct, 
and  that  either  method  will  give  the  same  result. 

In  Fig.  241,  a  similar  construction  is  shown.  The  points 
V  and  V7  are  the  vanishing  points  of  the  left  and  right  diagonals 
respectively.  The  perspective  a"  may  be  determined  by  the 
use  of  the  two  diagonals  without  the  aid  of  the  perpendicular. 
For  instance  ab  and  a'b'  are  corresponding  projections  of  one 
diagonal;  ad  and  a'd'  are  corresponding  projections  of  the  other 
diagonal.  Hence  W  is  the  indefinite  perspective  of  the  right 
diagonal  (since  cv'  is  drawn  to  the  right)  and  d'V  is  the  indefinite 
perspective  of  the  left  diagonal.  Their  intersection  determines 
a",  the  required  perspective  of  the  point  A.  The  indefinite  per- 
spective of  the  perpendicular  is  shown  as  a'c'  and  passes 
through  the  point  a"  as  it  should. 


248 


CONVERGENT  PROJECTING-LINE  DRAWING 


Any  two  lines  may  be  used  to  determine  the  perspective  and 
should  be  chosen  so  as  to  intersect  as  at  nearly  a  right  angle  as 
possible  to  insure  accuracy  of  the  location  of  the  point.  The 
visual  ray  may  also  be  drawn  on  this  diagram  and  its  piercing 
point  will  again  determine  a". 

1321.  To  find  the  perspective  of  a  line  by  the  method 
of  perpendiculars  and  diagonals.  Suppose  it  is  desired  to 
construct  the  perspective  of  an  arrow  by  the  method  of  perpendic- 
ulars and  diagonals.  Fig.  242  shows  a  case  of  this  kind.  AB 

is  an  arrow  situated  in  the  second 
angle;  C  is  the  point  of  sight. 
A  perpendicular  through  the 
point  of  sight  pierces  the  picture 
plane  in  the  vertical  projection  c'. 
The  two  possible  diagonals  whose 
horizontal  projections  are  given 
by  cm  and  en  pierce  the  picture 
plane  in  V  and  V  respectively. 
FIG.  242.-  The  perspective  of  any  diagonal 

drawn    through    any    point    in 

space  will  vanish  in  either  of  these  vanishing  points  depending 
upon  whether  the  diagonal  is  drawn  to  the  right  or  to  the  left 
of  the  point  in  question.  Likewise,  the  perspective  of  a  per- 
pendicular drawn  through  any  point  in  space  will  vanish  in  the 
centre  of  the  picture. 

In  the  case  at  issue,  the  horizontal  projection  of  a  diagonal 
through  b  is  bo  and  its  vertical  projection  is  b'o';  the  piercing 
points  is  at  o'.  As  the  perspectives  all  lines  parallel  to  the 
diagonal  have  a  common  vanishing  point,  then  the  perspective 
of  the  diagonal  through  B  must  vanish  at  V  and  the  perspective 
of  the  point  B  must  be  somewhere  on  the  line  joining  o'  and  V. 
Turning  attention  to  the  perpendicular  through  B,  it  is  found 
that  its  vertical  projection  corresponds  with  the  vertical  pro- 
jection of  the  point  itself  and  is  therefore  b'.  The  perspectives 
of  all  perpendiculars  to  the  vertical  or  picture  plane  vanish  in 
the  centre  of  the  picture;  and,  hence,  the  perspective  of  B  in 
space  must  be  somewhere  on  the  line  b'c'.  It  must  also  be 
somewhere  on  the  perspective  of  the  diagonal  and,  hence,  it  is 
at  their  intersection  b".  This  can  also  be  shown  by  drawing 


PERSPECTIVE  PROJECTION 


249 


the  visual  ray  through  the  point  of  sight  C  and  the  point  B. 
The  horizontal  and  vertical  projections  of  the  visual  ray  are 
cb  and  c'b'  and  they  pierce  the  picture  plane  at  b",  which  is  the 
same  point  obtained  by  finding  the  indefinite  perspectives  of 
the  perpendicular  and  the  diagonal. 

By  similar  reasoning  the  perspective  B!'  is  obtained  and, 
therefore,  a"b"  is  the  perspective  of  AB. 

1322.  Revolution  of  the  horizontal  plane.  The  fact  that 
the  second  and  fourth  angles  are  not  used  in  drawing  on  account 
of  the  conflict  between  the  separate  views  has  already  been  con- 
sidered (315,  317).  This  is  again  shown  in  Fig.  238.  The  two 
views  of  the  cube  overlap  and  make  deciphering  more  difficult. 
To  overcome  this  difficulty  the  horizontal  plane  is  revolved 

-f 


V 

A                    Horizon 

\—~  -^_^ 

4\0' 

\  \   i 

b< 

1 

y  ! 

\\ 

FIG.  243. 

180°  from  its  present  position.  This  brings  the  horizontal 
projections  below  the  ground  line  and  leaves  the  vertical  pro- 
jections the  same  as  before.  Now  the  diagonals  through  any 
point  slope  in  the  reverse  direction,  and  care  must  therefore  be 
used  in  selecting  the  proper  vanishing  point  while  drawing  the 
indefinite  perspective  of  the  diagonal. 

1323.  To  find  the  perspective  of  a  point  when  the  hori= 
zontal  plane  is  revolved.  Let  Fig.  243  represent  the  conditions 
of  the  problem.  The  vertical  projection  of  the  given  point  is 
a'  and  its  corresponding  horizontal  projection  is  at  a,  below  the 
ground  line  due  to  the  180°  revolution  of  the  horizontal  plane. 
The  vertical  projection  of  the  point  of  sight  is  at  c'  and  its  cor- 
responding horizontal  projection  is  at  c,  now  above  the  ground 
line.  The  conditions  are  such  that  the  given  point  seems  like 
a  first  angle  projection  and  the  point  of  sight  seems  like  a  second 


250 


CONVERGENT  PROJECTING-LINE  DRAWING 


angle  projection,  whereas  the  actual  conditions  are  just  the  reverse 
of  this. 

Through  a,  draw  ab  the  horizontal  projection  of  the  diagonal; 
the  corresponding  vertical  projection  is  a'b'  with  V  as  the  piercing 
point.  As  ab  is  drawn  to  the  right,  the  indefinite  perspective 
of  the  diagonal  must  vanish  in  the  left  vanishing  point  at  V, 
Hence,  draw  Vb',  the  indefinite  perspective  of  the  diagonal. 


y 

1 

y 

FIG.  243. 


The  indefinite  perspective  of  the  perpendicular  remains  unchanged 
and  is  shown  in  the  diagram  as  a'c'.  Therefore  the  intersection 
of  Vb'  and  a'c'  determine  a",  the  required  perspective  of  the  point 
A.  The  accuracy  of  the  construction  is  checked  by  drawing  the 
visual  ray  ca  and  then  erecting  a  perpendicular  at  o,  as  shown, 
which  passes  through  a"  as  it  should. 

1324.  To  find  the  perspective  of  a  line  when  the  hori- 
zontal plane  is  revolved.    The  point  B  in  space  will  again  be 

located  in  Fig.  244  just  as  was 
done  in  Fig.  242.  The  horizontal 
projection  of  the  diagonal  is  bo 
and  its  vertical  projection  is  b'o'; 
the  piercing  point,  therefore,  is 
o'.  The  horizontal  projection  of 
the  point  of  sight  c  is  now  above 
the  ground  line  instead  of  below 
it,  due  to  the  180°  revolution  of 
FlG  244  the  horizontal  plane;  the  orig- 

inal   position    of    this    point,  is 
marked  c".    A  diagonal  through  the  point  of  sight,  parallel  to 


PERSPECTIVE  PROJECTION  251 

the  diagonal  through  the  point  B,  is  shown  as  en  and  pierces 
the  vertical  plane  at  V  and,  therefore,  o'V  is  the  indefinite  per- 
spective of  the  diagonal.  The  indefinite  perspective  of  the 
perpendicular,  as  before,  is  b'c',  vanishing  in  the  centre  of  the 
picture.  The  intersection  of  these  two  indefinite  perspectives 
determines  b",  the  perspective  of  the  point  B  in  space. 

Instead  of  using  the  left  diagonal  bo,  through  b,  it  is  possible 
to  use  the  right  diagonal  bq,  and  this  pierces  the  picture  plane 
at  q',  and  vanishes  at  V  which  (as  must  always  be  the  case) 
again  determines  the  perspective  b".  It  must  be  observed  that 
it  matters  little  which  diagonal  is  used  with  a  perpendicular, 
or,  whether  only  the  diagonals  without  the  perpendicular  are 
used.  In  practice  such  lines  are  selected  as  will  intersect  as 
nearly  as  possible  at  right  angles  since  the  point  of  intersec- 
tion is  thereby  more  accurately  determined  than  if  the  two  lines 
intersected  acutely.  Whatever  is  done  the  principal  points 
can  always  be  located  by  any  two  lines,  and  the  other  may  be  used 
as  a  check.  Still  another  check  can  be  had  by  drawing  the  hori- 
zontal projection  of  the  visual  ray  cb  whereby  b",  the  perspective, 
is  once  more  determined.  The  point  A  in  space  is  located  in  an 
identical  manner. 

1325.  Location  of  diagonal  vanishing  points.    On  observa- 
tion of  Figs.  242  and  244,  it  is  seen  that  the  distance  V  to  c'  and 
V  to  c'  is  the  same  as  the  distance  of  the  point  of  sight  is  from 
the  vertical  plane.     This  is  true  because  a  45°  diagonal  is  used 
and  these  distances  are  the  equal  sides  of  a  triangle  so  formed. 
The  use  of  any  other  angle  would  not  give  the  same  result,  although 
the  distance  of  the  vanishing  points  from  the  centre  of  the  pic- 
ture would  always  be  equal. 

All  the  principles  that  are  necessary  for  the  drawing  of  any  kind 
of  a  linear  perspective  have  now  been  developed.  The  subsequent 
problems  will  illustrate  their  uses  in  a  variety  of  cases.  Certain 
adaptations  required  for  commercial  application  will  appear 
subsequently. 

1326.  Problem  2.    To  find  the  perspective  of  a  cube  by  the 
method  of  perpendiculars  and  diagonals. 

Construction.  Let  ABCDEFGH  in  Fig.  245  represent  the 
cube.  A  case  has  been  selected  that  is  identical  to  the  one 
shown  in  Fig.  238,  in  order  that  the  difference  between  the  two 


252 


CONVERGENT  PROJECTING-LINE  DRAWING 


methods  may  be  clearly  illustrated.  Suppose  it  is  desired  to 
locate  the  perspective  of  the  point  E.  Draw  from  e,  in  the  hori- 
zontal projection,  the  diagonal  eo,  the  vertical  projection  is  e'o' 
with  o'  as  a  piercing  point.  Join  o'  with  v,  and  the  indefinite 
perspective  of  the  diagonal  is  obtained.  The  perpendicular 
from  e  pierces  the  vertical  plane  at  e'  and  e's'  is  the  indefinite 
perspective.  The  intersection  E  of  these  indefinite  perspectives 
is  the  required  perspective  of  the  point.  This  point  can  also  be 
checked  by  drawing  the  other  diagonal  which  pierces  the  picture 
plane  at  p',  thereby  making  pV  the  indefinite  perspective  of 
the  latter  diagonal,  which  passes  through  E,  as  it  should. 

The  point  C  is  located  by  drawing  the  perpendicular  and 


FIG.  245. 

diagonal  in  the  same  way  as  was  done  for  the  point  E.  The 
rest  of  the  construction  has  been  omitted  for  the  sake  of  clearness. 
As  in  Fig.  242,  the  vanishing  points  V  and  V  are  laid  out  by 
drawing  a  line  through  the  point  of  sight  parallel  to  the  system 
of  lines.  Fig.  245  does  not  show  the  construction  in  this  way 
because  the  horizontal  projection  of  the  point  of  sight  is  now 
above  the  ground  line,  due  to  the  1 80  °-re volution  of  the  horizon- 
tal plane.  Instead,  the  lines  have  been  drawn  from  the  original 
horizontal  projection  shown  as  s"  and  care  has  been  taken  to  see 
that  the  lines  are  parallel  to  the  reversed  position  of  the  horizon- 
tal projection.  A  reference  to  both  Figs.  238  and  245  will  make 
this  all  clear. 

In  Fig.  245  two  vanishing  points  v'  and  v  will  be  noted  which 


PERSPECTIVE  PROJECTION 


253 


are  (as  has  already  been  shown  to  be)  the  vanishing  points  of  the 
perspectives  of  all  diagonals.  It  is  also  known  that  the  centre 
of  the  picture  s'  is  the  vanishing  point  of  the  perspectives  of  all 
perpendiculars.  Altogether,  there  are  five  vanishing  points; 
the  points  V  and  V  are  the  vanishing  points  of  the  perspectives 
of  the  horizontal  lines  on  the  cube. 

1327.  Problem  3.   To  find  the  perspective  of  a  hexagonal  prism. 

Construction.  Reference  to  Fig.  246  shows  that  one  edge 
of  the  prism  lies  in  the  picture  plane  while  the  base  is  in  the  hor- 
izontal plane.*  The  diagonals  used  to  determine  the  vanishing 
points  are  here  chosen  to  be  30°,  as  in  this  way  they  are  also 
parallel  to  some  of  the  lines  of  the  object  and,  therefore,  have 


FIG.  246. 

common  vanishing  points.  V'  and  V  are  the  vanishing  points 
of  the  diagonals,  and  of  those  sides  which  make  an  angle  of  30° 
with  the  picture  plane.  The  centre  of  the  picture  is  at  s'  and, 
therefore,  the  vanishing  point  of  the  perspectives  of  all  perpendic- 
ulars. The  edge  AG  lies  in  the  picture  plane,  and,  therefore, 
is  shown  in  its  actual  size.  The  indefinite  perspectives  of  the 
diagonals  from  the  extremities  A  and  G  will  give  the  direction 
of  the  sides  FA  and  AB  for  the  top  face  and  LG  and  GH  for  the 
base.  The  points  F  and  L  are  located  by  drawing  a  diagonal 
and  perpendicular  through  f  and  1  in  the  horizontal  projection. 
The  distance  mm'  and  nn'  is  equal  to  AG,  the  height  of  the  prism. 

*  This  picture  has  a  strained  appearance  due  to  the  selection  of  the  point 
of  sight.     See  Art.  1331  in  this  connection. 


254 


CONVERGENT  PROJECTING-LINE  DRAWING 


1328.  Problem  4.  To  find  the  perspective  of  a  pyramid 
superimposed  on  a  square  base. 

Construction.  In  Fig.  247,  the  edge  GK  is  shown  in  the 
picture  plane,  a  fact  which  enables  the  immediate  determination 
of  the  direction  of  JK,  KL,  FG  and  GH  by  joining  G  and  K  with 
the  vanishing  points  V'  and  V.  It  is  only  necessary  to  show  the 
points  J  and  L  to  determine  the  verticals  FJ  and  HL.  The 
method  for  this  has  already  been  shown.  One  other  important 
point  to  locate  is  the  apex  A.  This  is  shown  by  drawing  a  diagonal 
and  perpendicular  through  a.  The  line  AO,  in  space,  pierces  the 


picture  plane  at  o',  and  AP  at  p'  at  a  height  oo'  equal  to  the  height 
of  the  apex  above  the  horizontal  plane.  Joining  p'  with  s',  the 
indefinite  perspective  of  the  perpendicular  is  obtained.  At  its  inter- 
section with  the  indefinite  perspective  of  the  diagonal  oa,  locate 
A,  the  perspective  of  the  apex.  The  horizontal  projection  of 
the  point  of  sight  is  not  shown  as  the  location  of  v'  from  s'  and  v 
from  s'  also  gives  the  distance  of  the  point  of  sight  from  the  pic- 
ture plane  (1325). 

1329.  Problem  5.    To  find  the  perspective  of  an  arch. 

Construction.  Fig.  248  shows  plan  and  elevation  of  the 
arch.  The  corner  nearest  to  the  observer  is  in  the  picture  plane, 
and  therefore  the  lines  in  the  plane  are  shown  in  their  true  length. 


PERSPECTIVE  PROJECTION 


255 


256  CONVERGENT  PROJECTING-LINE  DRAWING 

This  length  need  not  be  in  actual  size  but  may  be  drawn  to  any 
scale  desired.  From  the  extremities  of  these  lines,  others  are 
drawn  to  the  vanishing  points  V  and  V,  as  these  are  the  van- 
ishing points  of  the  perspectives  of  the  parallels  to  the  sides  of 
the  arch. 

It  is  to  be  remembered  that  through  s,  a  line  sm  is  drawn 
parallel  to  the  reversed  direction  of  op,  because  the  plan  would 
ordinarily  be  above  the  ground  line.  This  convention  may  or 
may  not  be  adopted.  In  Fig.  249  it  is  shown  in  another  way 
and  perhaps  this  may  seem  preferable.  The  matter  is  one  of 
personal  choice,  however,  and  to  the  experienced,  the  liability  to 
error  is  negligible. 

The  most  important  feature  of  this  problem  is  the  location  of 
points  on  the  arch.*  The  advantage  of  the  use  of  perpendiculars 
and  diagonals,  to  locate  the  points  A,  B  and  C,  may  be  shown 
here.  Were  these  points  located  by  finding  the  piercing  points 
of  the  visual  ray,  the  reader  would  soon  find  the  number  of  lines 
most  confusing,  due  to  the  lack  of  symmetry  of  direction;  the 
use  of  the  45°  and  60°  triangles  for  the  diagonals  is  much  more 
convenient. 

In  commercial  perspective  drawing,  the  draftsman  estimates 
such  small  curves  as  are  shown  at  the  base.  Only  such  points 
are  located  which  are  necessary  for  guides.  In  this  case, 
the  verticals  and  their  limiting  position  are  all  that  are 
required. 

1330.  Problem  6.    To  find  the  perspective   of  a  building. 

Construction.  The  plan  and  elevations  of  the  building  are 
shown  on  Fig.  249  drawn  to  scale.  In  drawing  the  perspective, 
the  plan  only  is  necessary  to  locate  the  various  lines.  Continual 
reference  must  be  made  to  the  elevations  for  various  points  in 
the  height  of  the  building.  Considerable  work  can  be  saved  by 
having  one  corner  of  the  building  in  the  picture  plane.  Thus, 
that  corner  is  shown  in  its  true  length  even  though  it  may 
not  be  the  actual  length.  In  the  case  shown,  it  is  drawn  to 
scale. 

The  vanishing  points  of  the  horizontal  lines  on  the  building 
have  been  located  from  the  horizontal  projection  of  the  point 

*  This  arch  may  also  be  drawn  by  craticulation.     See  Art.  1331. 


PERSPECTIVE  PROJECTION 


257 


258  CONVERGENT  PROJECTING-LINE  DRAWING 

of  sight,  above  the  ground  line.  The  construction  lines  are 
therefore  drawn  parallel  to  the  sides  of  the  building.  The  dis- 
tances of  the  window  lines  may  be  laid  out  to  scale  on  the  corner 
of  the  building  which  lies  in  the  picture  plane,  as  shown.  Strictly 
speaking,  the  term  "  scale  "  cannot  be  applied  to  perspective 
drawing,  as  a  line  of  a  given  length  is  projected  as  a  shorter  line 
as  the  distance  from  the  observer  increases. 

1331.  Commercial  application  of  perspective.  The  artistic 
requirement  of  perspective  involves  some  choice  in  the  selection 
of  the  point  of  sight.  In  general,  the  average  observer's  point 
of  sight  is  about  five  feet  above  the  ground,  unless  there  is  some 
reason  to  change  it.  The  center  of  the  picture  should  be  chosen 
so  that  it  is  as  nearly  as  possible  in  the  centre  of  gravity  of  area 
of  the  picture  and  this  may  modify  the  selection  of  the  observer's 
point  of  sight.  For  instance,  if  a  perspective  of  a  sphere  is 
made  and  the  centre  of  the  picture  is  chosen  in  the  centre  of 
gravity  of  the  area,  then  the  perspective  is  a  circle.  Otherwise, 
if  the  centre  of  the  picture  is  to  one  side,  the  perspective  is  an 
ellipse,*  and,  to  properly  view  the  picture,  it  should  be  held  to 
one  side.  This  is  contrary  to  the  usual  custom;  an  observer 
holds  the  picture  before  him  so  that  the  average  of  the  visual 
rays  is  normal  to  the  plane  of  the  paper.  On  building  or  similar 
work,  the  roof  and  base  lines  should  not  converge  to  an  angle 
greater  than  about  50°.  This  can  be  overcome  by  increasing 
the  distance  between  the  vanishing  points,  or,  what  amounts  to 
the  same  thing,  increasing  the  distance  between  observer  and 
object. 

It  is  also  desirable  to  choose  the  position  of  the  observer  so  as 
to  show  the  most  attractive  view  of  the  object  most  prominently. 
Where  a  building  has  two  equally  prominent  adjacent  sides, 
it  is  a  good  plan  to  adjust  the  observer's  position  to  present  the 
sides  approximately  in  proportion  to  their  lengths.  That  is, 
when  one  side  of  a  building  is  200  feet  long  and  50  feet  wide, 
the  length  of  the  building  should  appear  on  the  drawing  about 
four  times  as  long  as  the  width  (these  recommendations  have 
been  purposely  ignored  in  Fig.  249). 

*  The  limiting  rays  from  the  sphere  are  elements  of  the  surface  of  a 
cone  of  revolution;  hence,  its  intersection  with  a  plane  inclined  to  its  axis 
will  be  an  ellipse. 


PERSPECTIVE  PROJECTION 


259 


When  impressions  of  magnitudes  of  objects  are  to  be  con- 
veyed, it  can  be  done  by  placing  men  at  various  places  on  the 
drawing.  A  mental  estimate  of  the  height  of  a  man  will  roughly 
give  an  estimate  of  the  magnitude  of  the  object. 

When  curves  are  present  in  a  drawing,  the  use  of  bounding 
figures  of  simple  shape  again  finds  application  (210,  408). 
Architects  know  this  under  the  name  of  craticulation.  Fig,  250 


shows  an  example  as  applied  to  the  drawing  of  a  gas-engine 
fly-wheel.  A  slightly  better  picture  could  have  been  obtained 
by  giving  the  wheel  a  tilt  so  as  to  show  the  "  section  "  more 
prominently.  This  would  also  bring  the  centre  of  the  picture 
nearer  to  the  centre  of  gravity  of  the  area.  If  the  reader  will 
study  a  few  photographs,  keeping  these  suggestions  in  mind  he 
will  note  the  conditions  which  make  certain  pictures  better  than 
others. 


260  CONVERGENT  PROJECTING-LINE  DRAWING 

To  make  large  perspectives  is  sometimes  inconvenient  due 
to  the  remoteness  of  the  vanishing  points.  This  requires  a  large 
drawing  board  and  frequently  the  accuracy  is  impaired  by  the 
unwieldiness  of  the  necessary  drawing  instruments.  To  over- 
come these  objections,  it  is  possible  to  make  a  small  drawing  of 
the  object  and  then  to  redraw  to  a  larger  one  by  means  of  a  pro- 
portional divider  or  by  a  pantograph.  Only  the  more  important 
lines  need  be  located  on  the  drawing.  Such  details  as  windows, 
doors,  etc.,  can  usually  be  estimated  well  enough  so  as  to  avoid 
suspicion  as  to  their  accuracy.  The  artist,  with  a  little  practice, 
soon  accustoms  himself  to  filling  in  detail. 

Freehand  perspective  sketches  can  be  made  with  a  little 
practice.  To  make  these,  lay  out  the  horizon  and  two  vanish- 
ing points  for  building  work  and  perhaps  one  or  more  for  machine 
details.  Accuracy  is  usually  not  a  prerequisite,  and,  therefore, 
sketches  of  this  kind  can  be  made  in  almost  the  same  time  as 
oblique  or  axonometric  sketches.  The  experience  gained  from 
this  chapter  should  have  furnished  sufficient  principles  to  be 
applied  directly. 

The  chief  function  of  perspective  is  to  present  pictures  to 
those  unfamiliar  with  drawing.  It  is  therefore  desirable  to  use 
every  effort  possible  to  present  the  best  possible  picture  from  the 
artistic  viewpoint.  The  increased  time  required  to  make  per- 
spectives is  largely  offset  by  the  ease  with  which  the  uninitiated 
are  able  to  read  them.  The  largest  application  of  perspective 
is  found  in  architect's  drawings  to  clients,  and  in  the  making  of 
artistic  catalogue  cuts.  With  perspective  is  usually  associated 
the  pictorial  effect  of  illumination  so  as  to  call  on  the  imagination 
to  the  minimum  extent. 

1332.  Classification  of  projections.*  When  drawings  are 
made  on  a  plane  surface  then  there  are  two  general  systems 
employed:  those  in  which  the  projecting  lines  are  parallel  to 
each  other  and  those  in  which  the  projecting  lines  converge  to  a 
point. 

*  See  Art.  413  in  this  connection. 


PERSPECTIVE  PROJECTION 


261 


Commonly 

called  ortho- 

Showing two 

graphic  pro- 

dimensions 

jections    or 

Mechanical 

drawings. 

Orthographic  • 

Showing  three 
dimensions 

Isometric 

'  Parallel 
projecting 

and  known 
as  axonome- 

Dimetric 

lines 

tric     projec- 

Trimetric 

Projections 
on  plane        " 

Oblique 

tions. 

surfaces 

Convergent 
projecting      . 

\  Outline  alone  is  shown 
perspective     I 

lines 

Aerial                (  Color  and  illumination  added 

perspective    I    to  linear  perspective 

QUESTIONS  ON  CHAPTER  XIII 

1.  What  is  a  scenographic  projection? 

2.  What  is  a  linear  perspective? 

3.  Define  visual  rays. 

4.  Define  visual  angle. 

5.  What  is  the  physiological  effect  of  a  variation  in  the  visual  angle? 

6.  How  does  binocular  vision  afford  a  means  of  estimating  distance? 

7.  Why  is  the  eye  assumed  as  a  single  point  in  perspective? 

8.  What  is  a  vanishing  point?     Give  example. 

9.  Is  the  vanishing  point  the  vanishing  point  of  the  line  or  of  its  per- 

spective? 

10.  Show  the  theory  of  perspective  by  a  window-pane  illustration. 

11.  What  is  an  aerial  perspective? 

12.  What  is  the  picture  plane? 

13.  How  is  the  picture  plane  usually  located  with  respect  to  the  object 

and  the  observer? 

14.  Show  by  an  oblique  projection  how  the  perspective  of  a  line  is 

constructed. 

15.  Show  how  the  perspective  of  a  line  is  constructed  in  orthographic 

projection. 

16.  Prove  that  all  perspectives  of  lines  perpendicular  to  the  horizontal 

plane  have  vertical  perspectives. 

17.  Prove  that  all  perspectives  of  lines  parallel  to  both  principal  planes 

have  perspectives  parallel  to  the  ground  line. 

18.  What  is  the  centre  of  the  picture? 

19.  Prove  that  the  perspectives  of  all  lines  perpendicular  to  the  picture 

plane  vanish  in  the  centre  of  the  picture. 


262  CONVERGENT  PROJECTING-LINE  DRAWING 

20.  Is  the  centre  of  the  picture  the  vertical  projection  of  the  point  of  sight? 

21.  What  is  a  system  of  lines? 

22.  Prove  that  the  perspectives  of  all  systems  of  lines  have  a  common 

vanishing  point. 

23.  How  is  the  vanishing  point  of  a  system  of  lines  found? 

24.  What  is  the  horizon? 

25.  Prove  that  the  perspectives  of  all  horizontal  systems  of  lines  have 

a  vanishing  point  on  the  horizon. 

26.  Why  is  the  centre  of  the  picture  on  the  horizon? 

27.  Find  the  perspective  of  a  point  which  is  located  in  the  second  angle, 

by  means  of  the  piercing  point  of  the  visual  ray  on  the  picture 
plane. 

28.  Find  the  perspective  of  a  point  which  is  located  in  the  first  angle, 

by  means  of  the  piercing  point  of  the  visual  ray  on  the  picture 
plane. 

29.  Show  that  the  vertical  projection  of  a  point  which  is  situated  in 

the  picture  plane  is  its  own  perspective. 

30.  Show  that  the  vertical  projection  of  a  line  which  is  situated  in  the 

picture  plane  is  its  own  perspective. 

31.  What  is  the  indefinite  perspective  of  a  line?    Give  proof. 

32.  Find  the  perspective  of  a  cube  by  means  of  the  piercing  points'of 

the  visual  rays  on  the  picture  plane. 

33.  Prove  that  if  two  lines  in  space  intersect,  their  perspectives  intersect 

in  a  point  which  is  the  perspective  of  the  point  of  intersection 
on  the  lines. 

34.  What  is  a  perpendicular  when  applied  to  perspective? 

35.  Where  does  the  perspective  of  a  perpendicular  vanish? 

36.  What  is  a  diagonal  when  applied  to  perspective? 

37.  How  many  diagonals  may  be  drawn  through  a  given  point? 

38.  Where  do  the  diagonals  vanish?    Why? 

39.  What  angle  is  generally  used  for  the  diagonals?    What  other  angles 

may  be  used? 

40.  Find  the  perspective  of  a  second  angle  point,  by  the  method  of 

perpendiculars  and  diagonals. 

41.  Find  the  perspective  of  a  second  angle  point,  by  drawing  two  diagonals 

through  it.     Check  by  drawing  the  perpendicular. 

42.  Find  the  perspective  of  a  first  angle  point,  by  the  method  of  per- 

pendiculars and  diagonals. 

43.  Find  the  perspective  of  a  first  angle  point,  by  drawing  two  diagonals 

through  it.     Check  by  drawing  the  perpendicular. 

44.  Find  the  perspective  of  a  second  angle  line,  by  the  method  of  per- 

pendiculars and  diagonals. 

45.  Find  the  perspective  of  a  second  angle  line,  by  drawing  the  diagonals 

only.     Check  by  drawing  the  perpendicular  and  visual  rays. 

46.  Find  the  perspective  of  a  first  angle  line,  by  the  method  of  perpen- 

diculars and  diagonals. 

47.  Find  the  perspective  of  a  first  angle  line,  by  drawing  the  diagonals 

only.    Check  by  drawing  the  perpendiculars  and  visual  rays. 


PERSPECTIVE  PROJECTION  263 

48.  What  is  the  object  of  revolving  the  horizontal  plane  in  perspective? 

49.  What  precaution  must  be  used  in  perspective  with  reference  to  the 

diagonals,  when  the  horizontal  plane  is  revolved? 

50.  Find  the  perspective  of  a  second  angle  point,  by  the  method  of 

perpendiculars  and  diagonals,  when  the  horizontal  plane  is  revolved. 

51.  Find  the  perspective  of  a  first  angle  point,  by  the  method  of  per- 

pendiculars and  diagonals,  when  the  horizontal  plane  is  revolved. 

52.  Find  the  perspective  of  a  second  angle  line,  by  the  method  of  per- 

pendiculars and  diagonals,  when  the  horizontal  plane  is  revolved. 

53.  Find  the  perspective  of  a  first  angle  line,  by  the  method  of  perpen- 

diculars and  diagonals,  when  the  horizontal  plane  is  revolved. 

54.  When  the  diagonals  make  an  angle  of  45°  with  the  picture  plane, 

how  far  are  the  vanishing  points  from  the  centre  of  the  picture? 

55.  Find  the  perspective  of  a  cube  by  means  of  perpendiculars  and 

diagonals. 

56.  Why  should  the  centre  of  the  picture  be  chosen  as  near  as  possible 

to  the  centre  of  gravity  of  the  perspective? 

57.  What  is  the  perspective  of  a  sphere,  when  the  centre  of  the  picture 

is  in  the  centre  of  gravity  of  the  perspective? 

58.  What  is  the  perspective  of  a  sphere,  when  the  centre  of  the  picture 

does  not  coincide  with  the  centre  of  gravity  of  the  perspective? 

59.  What  should  be  the  approximate  angle  of  convergence  of  the  roof 

and  base  lines  of  a  building  on  a  perspective? 

60.  If  the  roof  and  base  lines  of  a  building  converge  to  an  angle  con- 

sidered too  large,  what  remedy  is  there  for  this  condition? 

61.  When  adjacent  sides  of  a  building  are  equally  attractive,  what  should 

be  their  general  relation  on  the  perspective? 

62.  How  may  impressions  of  magnitude  be  conveyed  on  a  perspective? 

63.  What  is  craticulation? 

64.  How  may  large  perspectives  be  made  when  the  drawing  board  is 

small? 

65.  When  perspective  is  commercially  applied,  is  it  necessary  to  locate 

every  point  on  the  details  or  can  it  be  drawn  with  sufficient 
accuracy  by  estimation? 

66.  Make  a  complete  classification  of  projections  having  parallel  and 

convergent  projecting  lines. 

Note.  In  the  following  drawings,  keep  the  centre  of  the  picture  as 
near  as  possible  to  the  centre  of  gravity  of  the  drawing. 

67.  Make  a  perspective  of  Fig.  1  in  text. 

68.  Make  a  perspective  of  Fig.  10  in  text. 

69.  Make  a  perspective  of  Fig.  17  in  text. 

70.  Make  a  perspective  of  Fig.  2A.     (Question  in  Chap.  II.) 

71.  Make  a  perspective  of  Fig.  2B.     (Question  in  Chap.  II.) 

72.  Make  a  perspective  of  Fig.  21  in  text. 

73.  Make  a  perspective  of  Fig.  3 A.     (Question  in  Chap.  III.) 

74.  Make  a  perspective  of  Fig.  3B.  "  " 

75.  Make  a  perspective  of  Fig.  3C.  "        "  " 


264  CONVERGENT  PROJECTING-LINE  DRAWING 

76.  Make  a  perspective  of  Fig.  3D.     (Question  in  Chap.  III.) 

77.  Make  a  perspective  of  Fig.  3E. 

78.  Make  a  perspective  of  Fig.  3F. 

79.  Make  a  perspective  of  Fig.  3G. 

80.  Make  a  perspective  of  Fig.  3H. 

81.  Make  a  perspective  of  Fig.  37. 

82.  Make  a  perspective  of  Fig.  3-7. 

83.  Make  a  perspective  of  Fig.  3K. 

84.  Make  a  perspective  of  Fig.  3L. 

85.  Make  a  perspective  of  Fig.  40  in  text. 

86.  Make  a  perspective  of  Fig.  41  in  text. 

87.  Make  a  perspective  of  Fig.  42  in  text. 

88.  Make  a  perspective  of  a  pyramid  on  a  square  base. 

89.  Make  a  perspective  of  an  arch. 

90.  Make  a  perspective  of  a  building. 


PART  IV 
PICTORIAL  EFFECTS  OF  ILLUMINATION 


CHAPTER  XIV 

PICTORIAL    EFFECTS    OF  ILLUMINATION  IN  ORTHOGRAPHIC 

PROJECTION 

1401.  Introductory.  The  phenomena  of  illumination  on 
objects  in  space  is  a  branch  of  the  science  of  engineering  drawing,, 
which  aims  to  give  the  observer  a  correct  imitation  of  the  effect 
of  light  on  the  appearance  of  an  object.  In  the  main,  it  is  desired 
to  picture  reality.  The  underlying  principles  of  Rumination 
are  taken  from  that  branch  of  physics  known  as  Optics  or  Light. 
The  application  of  these  principles  to  graphical  presentation 
properly  forms  a  part  of  drawing. 

It  is  needless  to  say  that  the  subject  borders  on  the  artistic; 
yet  it  is  sometimes  desirable  to  present  pictures  to  those  who  are 
unfamiliar  with  the  reading  of  commercial  orthographic  pro- 
jections. The  architect  takes  advantage  of  perspective  in  con- 
nection with  the  effects  of  illumination,  and  in  this  way  brings 
out  striking  contrasts  and  forcibly  attracts  attention  to  the  idea 
expressed  in  the  drawing. 

It  is  not  always  essential,   however,   to  bring  out  striking 
effects,  as    occasion  often  arises  to  picture  the  surfaces  of  an 
object.     This    is    done    by   pic- 
torially  representing   the  effects 
of  illumination. 


1402.  Line  shading  applied 
to  straight  lines.  The  simplest 
application  of  line  shading  is 
shown  in  Fig.  251;  the  object 

is  a  rectangular  cover  for  a  box.     In  the  illumination,  it  is  assumed 

265 


FIG.  251. 


266 


PICTORIAL  EFFECTS   OF  ILLUMINATION 


FIG.  251. 


that  the  light  comes  in  parallel  lines,  from  the  upper  left-hand 
corner  of  the  drawing.  The  direction  is  downward,  and  to  the 

right,  at  an  angle  of  45°.  The 
illuminated  surfaces  are  drawn 
with  any  thickness  (or,  as  is 
commercially  termed,  "  weight  ") 
of  line;  the  surfaces  not  in  the 
light  are  made  heavier  as  though 
the  line  or  surface  actually  cast 
a  shadow.  Irrespective  of  the 

location  on  the  sheet,  the  drawing  would  always  be  represented 
in  the  same  way  since  the  light  is  assumed  to  come  in  parallel 
lines. 

Evidently  there  is  (strictly  speaking)  no  underlying  theory 
in  this  mode  of  shading,  the  process  being  merely  a  convention 
adopted  by  draftsmen,  thus  its  extended  use  merely  is  the  neces- 
sary recommendation  for  its  value. 

1403.  Line  shading  applied  to  curved  lines.     In  Fig.  252 


FIG.  252. 


is  shown  one  part  of  a  flange  coupling,  the  illustration  being 
chosen  to  show  the  application  of  line  shading  to  the  drawing  of 
curves.  In  this  latter  application  all  is  given  that  is  necessary 
to  make  drawings  using  this  mode  of  representat'on. 

As  before,  the  light  comes  from  the  upper   left-hand  corner 


IN  ORTHOGRAPHIC  PROJECTION  267 

of  the  drawing.  By  the  aid  of  the  two  views  (one-half  of  one 
being  shown  in  section)  such  surfaces  as  will  cast  a  shadow  are 
easily  distinguished.  The  shade  lines  of  the  circles  are  shown 
to  taper  gradually  off  to  a  diagonal  line  ab.  In  drawing  the  shade 
line  of  a  circle,  for  instance,  draw  the  circle  with  the  weight  of 
line  adopted  in  making  the  drawing;  with  the  same  radius  and 
a  centre  located  slightly  eccentric  (as  shown  much  exaggerated 
at  c  in  Fig.  252),  draw  another  semicircle,  adding  thickness  to 
one  side  of  the  circle  or  the  other,  depending  upon  whether  it  is 
illum'nated  or  not. 

It  will  be  seen  that  this  second  circle  will  intersect  the  first 
somewhere  near  the  diagonal  ab.  For  instance,  the  extreme 
outside  circle  casts  a  shadow  on  the  lower  right-hand  side,  whereas 
the  next  circle  is  shaded  on  the  upper  left-hand  side.  These 
two  shaded  circles  indicate  that  there  is  a  projection  on  the 
surface.  In  the  same  manner  the  drawing  is  completed.  It 
makes  no  difference  whether  the  surface  projects  much  or  little, 
the  lines  are  all  shaded  in  the  same  way  and  with  the  same  weight 
of  shade  line. 

Notice  should  be  taken  that  for  all  concentric  circles,  the 
eccentric  centre  is  always  the  same.  The  six  bolt  holes  are 
also  shown  shaded,  and  each  of  which  has  its  own  eccentric 
centre. 

Many  cases  arise  in  practice  where  there  are  projec- 
tions or  depressions  in  the  surface.  This  convention  helps  to 
interpret,  rapidly  and  correctly,  such  drawings.  The  time 
taken  to  use  this  method  is  more  than  offset  by  the  advantages 
thereby  derived;  only  in  extremely  simple  drawings  does  it  become 
unnecessary. 

1404.  Line  shading  applied  to  sections.     It  makes  no  dif- 
ference   whether   the    outside  view  is    shown,   or  whether  the 
object    is    shown    in   section    (Fig.   252)    the   shade    lines    are 
drawn    as    though    the   supplementary  planes   actually  cut  the 
object  so  as  to  expose  that  portion.     This  is  done  for  uniformity 
only. 

1405.  Line  shading  applied  to  convex  surfaces.     Occa- 
sionally, curved  surfaces  must  be  contrasted  with  flat  surfaces,  or, 
perhaps,  the  effect  of  curvature  brought  out  without  an  attempt  to 


268  PICTORIAL  EFFECTS  OF  ILLUMINATION 

contrast  it  with  flat  surfaces.  Fig.  253  shows  a  cylinder  shaded. 
The  heavier  shade  is  shown  to  the  right  as  the  light  is  supposed 
to  come  from  the  left;  the  reason  for  this  will  be  shown  later. 
The  effect  of  shading  and  its  graduation  is  produced  by  gradually 
altering  the  space  between  the  lines,  however,  keeping  the  weight 
of  line  the  same.  A  somewhat  better  effect  is  produced  by  also 
increasing  the  weight  of  the  line  in  connection  with  the  decrease 
in  the  spacing,  but  the  custom  is  not  general. 

1406.  Line  shading  applied  to  concave  surfaces.  A  hollow 
cylinder  is  shown  shaded  in  Fig.  254.  Again,  the  light  comes 
from  the  left  and  the  heavier  shade  falls  on  the  left,  as  shown. 


FIG.  253.  FIG.  254. 

A  comparison  between  Figs.  253  and  254  shows  how  concave  and 
convex  surfaces  can  be  indicated. 

1407.  Line    shading    applied    to    plane    surfaces.      Flat 
surfaces  are  sometimes  shaded  by  spacing  lines  an  equal  distance 
apart.     The  effect  produced  is  that  of  a  flat  shade.     The  method 
is  used  when  several  surfaces  are  shown  in  one  view  whose  planes 
make  different  angles  with  the  planes  of  projection,   like  an 
octagonal  prism,  for  instance. 

PHYSICAL  PRINCIPLES  OF  LIGHT . 

1408.  Physiological   effect   of    light.      Objects    are   made 
evident  to  us  by  the  reflected  light  they  send  to  our  eyes.     The 
brain  becomes  conscious  of  the  form  of  the  object  due  to  the 
physiological  effect  of  light  on  the  retina  of  the  eye.     The  locality 
from  which  the  light  emanates  is  called  the  source.     Light  travels 
in  straight  lines,  called  rays,  unless  obstructed  by  an  opaque 
body.     If  the  source  of  light  is  at  a  considerable  distance  from 
the  object,  the  light  can  be  assumed  to  travel  in  parallel  lines. 


IN  ORTHOGRAPHIC  PROJECTION  269 

The  chief  source  of  light  is  the  sun  and  its  distance  is  so  great 
(about  93,000,000  miles)  that  all  rays  are  practically  parallel. 

1409.  Conventional  direction  of  light  rays.    The  source 
of  light  may  be  located  anywhere,  but  it  is  usual  to  assume  that 
it  comes  from  over  the  left  shoulder  of  the  observer  who  is  viewing 
an  object  before  him.     The  projections  of  the  rays  on  the  hor- 
izontal and  vertical  plane  make  an  angle  of  45°  with  the  ground 
line.     The  problems  to  follow  will  be  worked  out  on  this  assumed 
basis. 

1410.  Shade  and  shadow.     The  part  of  the  object  that  is 
not  illuminated  by  direct  rays  is  termed  the  shade.     The  area 
from  which  light  is  excluded,  whether  on  the  object  itself  or  on 
any  other  surface,  is  called  the  shadow.     As  an  example,  take  a 
building  with  the  sun  on  one  side;   the  opposite  side  is  evidently 
in  the  shade.     A  cornice  on  this  building  may  cast  a  shadow 
on  the  walls  of  the  building,  while  the  entire  structure  casts  a 
shadow  on  the  ground,  and  sometimes  on  neighboring  buildings. 

1411.  Umbra  and  penumbra.     When  the   source  of   light 
is  chosen  near  the  object,  two  distinct  shadows  are  observable, 
one  within  the  other.     Fig.  255  shows  a  plan  view  of  a  flat  gas 
flame  ab  and  a  card  cd.     Rays  emanate  from  every  point  of  the 
flame  in  all   directions.     Consider  the  point  a,  in  the    flame. 
The  two  rays  ac  and  ad  will  determine  a 

shadow  cast  by  the  card  the  area  of  which 
is  located  away  from  the  flame  and  limited 
to  that  behind  fcdh.  Similarly,  from  the 
point  b,  the  two  rays  be  and  bd  cast  a  a 
shadow,  limited  by  the  area  behind  ecdg. 
These  two  areas  overlap.  From  the  posi- 
tion of  fcdg,  no  part  of  the  flame  can  be 
seen  by  an  observer  standing  there;  but 
from  any  one  of  the  areas  behind  eof  or  gdh, 
a  portion  of  the  flame  can  be  seen.  The 
effect  of  this  is  that  the  shadow  in  fcdg  is  much  more  pronounced 
and  therefore  is  the  darker. 

That  portion  of  the  shadow  from  which  the  light  is  totally 
excluded  (fcdg)  is  called  the  umbra;  that  portion  from  which 
the  light  is  partly  excluded  (ecf  or  gdh)  is  called  the  penumbra. 


270 


PICTORIAL  EFFECTS  OF  ILLUMINATION 


This  effect  is  easily  shown  by  experiment,  and  is  sometimes 
seen  in  photographs  taken  by  artificial  illumination  when  the 
source  of  light  is  quite  near  the  object.  In  passing,  it  may  also 
be  mentioned  that  if  there  are  two  sources  of  light,  there  may  be 
two  distinct  shadows,  each  having  its  own  umbra  and  penumbra, 
but  such  cases  are  not  usually  considered  in  drawing. 

When  applying  these  principles  to  drawing,  the  shadow 
is  limited  to  the  umbra,  and  the  penumbra  is  entirely  neglected. 
The  light  is  supposed  to  come  from  an  infinite  distance,  and, 
therefore,  in  parallel  rays.  So  little  in  the  total  effect  of  a  draw- 
ing is  lost  by  making  these  assumptions,  that  the  extra  labor 
involved  in  making  theoretically  correct  shadows  is  uncalled  for. 


GRAPHICAL  REPRESENTATION 

1412.  Application  of  the  physical  principles  of  light  to 
drawing.    In  the  application  of  the  physical  principles  enumerated 
above,  it  is  merely  necessary  to  draw  the  rays  of  light  to  the 
limiting  lines  of  the  object,  and  find  their  piercing  points  with 
the  surface  on  which  the  shadow  is  cast.     This  is  the  entire 
theory. 

1413.  Shadows  of  lines.*     Fig.  256"exhibits  the  shadow  cast 
by  an  arrow  AB  shown  by  its  horizontal  projection  ab  and  its 


FIG.  256. 


FIG.  257. 


vertical  projection  a'b'.     The  rays  of  light  come,  as  assumed, 

*  Lines  are  mathematical  concepts  and  have  no    width  or  thickness. 
They  can,  therefore,  fetrictly  speaking,  cast  no  shadow. 


IN  ORTHOGRAPHIC  PROJECTION  271 

in  lines  whose  horizontal  and  vertical  projections  make  an  angle 
of  45°  with  the  ground  line.  A  ray  of  light  to  A  is  shown  as  ca 
in  the  horizontal  projection  and  c'a'  in  the  vertical  projection. 
The  ray  of  light  to  B  is  shown  by  the  similar  projection  db 
and  d'b'.  These  two  rays  pierce  the  horizontal  plane  at  a?' 
and  b"  and  a"b"  is  therefore  the  shadow  of  the  arrow  AB  in 
space. 

If  the  arrow  is  as  shown  in  Fig.  257,  then  the  shadow  is  on  the 
vertical  plane.  A  similar  method  is  used  for  drawing  the  pro- 
jections of  the  rays  on  the  horizontal  and  vertical  planes.  Instead, 
however,  the  piercing  points  on  the  vertical  plane  are  found, 
and  these  are  shown  as  a"b",  which  again  is  the  shadow  of  AB 
in  space. 

In  Fig.  256  the  rays  pierce  the  horizontal  plane  before  they 
pierce  the  vertical  plane  and  the  shadow  is  therefore  on  the 
horizontal  plane.     Since  the  plane  is  assumed  opaque  there  is  no 
vertical  shadow.     Fig.   257  is  the  reverse  of  this.     A  vertical 
shadow  and  no  horizontal  shadow  is 
obtained.      Cases    may    occur   where 
the  shadow  is  partly  on  both  planes, 
similar  to  a  shadow  cast  partly  on  the 
floor  and  partly  on  the  wall  of  a  room.  — 
Fig.  258  shows  this  in  construction. 
It  differs  from  the  preceding  in  so  far 
as  the  two  horizontal  and  the  two  ver- 
tical piercing  points  of  the  rays  are  FIG.  258. 
obtained,  and  thus  the  direction  of  the 

shadow  is  determined  on  both  planes.  If  the  construction  is 
carried  out  accurately,  the  projections  will  intersect  at  the  ground 
line,  and  the  shadow  will  appear  shown  as  a"b". 

1414.  Problem  1.  To  find  the  shadow  cast  by  a  cube  which 
rests  on  a  plane. 

Let,  in  Fig.  259,  abed  be  the  horizontal  projection  of  the  top 
of  the  cube  and  a'b'c'd'  be  the  vertical  projection  of  the  top  of 
the  cube.  The  three  points  in  space  A,  B  and  C  will  be  sufficient 
to  determine  the  shadow  and,  as  a  consequence,  the  projections 
of  these  rays  are  drawn  an  angle  of  45°,  as  shown.  The  hor- 
izontal projections  of  the  rays  are  accordingly  ae,  bf  and  eg; 
the  vertical  projections  are  likewise  a'e',  b'f  and  c'g'.  The 


272 


PICTORIAL  EFFECTS  OF  ILLUMINATION 


piercing  points  (506,  805)  of  these  rays  are  e,  f  and  g.  The 
shadow  ir  therefore  determined  because  the  cube  rests  on  the 
horizontal  plane,  and  the  base  must  necessarily  be  its  own 
shadow. 

1415.  Problem  2.  To  find  the  shadow  cast,  by  a  pyramid, 
on  the  principal  planes. 

In  order  to  show  a  case  where  the  shadow  is  cast  on  both 
planes,  the  pyramid  must  be  located  close  to  the  vertical  plane. 
The  pyramid,  ABCDE,  Fig.  260,  is  shown  in  the  customary  way 


FIG.  259 


as  abcde  or  the  horizontal  projection,  and  a'b'c'd'e'  for  the 
vertical  projection.  The  horizontal  piercing  point  of  what  would 
be  the  shadow  of  the  apex  must  be  located  in  order  to  determine 
the  shadow  of  the  side  BA,  even  though  it  forms  no  part  of  the 
actual  shadow.  Locate  also  the  vertical  piercing  point  g'  which 
is  the  actual  shadow  of  the  apex.  It  will  be  noticed  that  the 
shadow  of  BA  only  continues  until  it  meets  the  vertical  plane  and 
is  therefore  limited  by  the  projection  bh.  Similarly,  the  shadow 
is  limited  by  di,  a  portion  of  the  line  df.  Where  these  two  lines 
meet  the  ground  line,  join  the  points  with  the  actual  shadow 
of  the  apex  and  the  shadow  is  then  completed.  It  may  be 


IN  ORTHOGRAPHIC  PROJECTION 


273 


observed  that  the  limiting  lines  of  an  object  determine  the 
shadow  in  all  cases,  and  that  the  interior  lines  do  not  influence 
the  figure  unless  they  project  above  the  rest  of  the  object  in  any 
way. 

1416.  Problem  3.  To  find  the  shade  and  shadow  cast  by 
an  octagonal  prism  having  a  superimposed  octagonal  cap. 

Let  Fig.  261  represent  the  object  in  question.  Confine  atten- 
tion at  present  to  the  shadow  on  the  horizontal  plane.  Con- 

c'b'        d'a'         e'fi        fg 


FIG.  261. 


sider,  first,  the  shadow  cast  by  the  line  GH  in  space.  It  will 
be  seen  that  the  rays  of  light  from  G  and  H  pierce  the  horizontal 
plane  at  points  g"  and  h"  and  that  these  points  are  found  by 
drawing  the  projections  of  the  rays  from  the  points  G  and  H  in 
space.  If  the  construction  is  correct  so  far,  then  g"h"  should 
be  parallel  to  gh,  as  GH  is  parallel  to  the  horizontal  plane  and  as 
its  shadow  must  be  parallel  both  to  the  line  itself  and  also  to 
its  horizontal  projection.  Similarly,  locate  a"  and  i" '.  Thus, 
three  lines  of  the  shadow  are  determined. 

It  is  now   necessary  to    observe  carefully  that  the  shadow  of 


274 


PICTORIAL  EFFECTS   OF  ILLUMINATION 


the  superimposed  cap  is  not  only  due  to  the  top  face,  as  the  points 
m"n"o"p"  are  the  piercing  points  of  rays  that  come  from  the 
points  MNOP,  in  space,  and  that  they  are  on  the  lower  face  of 
the  cap.  The  shadow  of  the  points  BCDE  fall  within  the  area 
a"h"g"f"p"o"m"n"  and  therefore  need  not  be  located.  The 
shadow  is  completed  by  drawing  r"s"  and  t"u"  which  are  a  part 
of  the  shadow  cast  by  the  octagonal  prism  below.  It  is  unneces- 
sary to  determine  the  Complete  outline  of  the  shadow  of  the 


c'b'       d'a!         e'ti       f'J 


mf: 


prism  as  it  merges  into  that  cast  by  the  cap.     The  shadow  on 
the  horizontal  plane  is  thus  completely  determined. 

Consider,  in  addition,  the  shadow  cast  on  the  prism  due  to 
the  superimposed  cap.  The  shadow  limited  by  the  line  v"w" 
is  due  to  the  limited  portion  of  the  lower  face  of  the  cap  shown 
horizontally  projected  at  vw  and  vertically  as  v'w'.  The  shadow 
of  the  point  O  in  space  is  shown  as  x"  and  that  of  Y  in  space  as 
y".  All  these  latter  points  are  joined  by  straight  lines  as  they 
are  shadows  of  straight  lines  cast  upon  a  plane  surface  and  may 
be  considered  as  the  intersection  of  a  plane  of  rays  through  the 
line  forming  the  outline  of  the  object. 


IN  ORTHOGRAPHIC  PROJECTION 


275 


The  right  of  the  face  of  the  prism  and  of  the  cap  is  shaded 
entirely,  because  it  is  in  the  shade,  and  since  it  receives  no  direct 
illumination,  it  is  represented  as  shown. 

1417.  Problem  4.    To   find   the    shade   and   shadow   cast 
by  a  superimposed  circular  cap  on  a  cylinder. 

Fig.  262  shows  the  cylinder  with  its  superimposed  cap.  The 
shadow  cast  by  the  cap  will  be  some  form  of  curve,  and  attention 
will  be  directed  at  present,  to  the  location  points  on  this  curve. 
From  a',  in  the  vertical  projection,  draw  a  ray  whose  projection 
makes  an  angle  of  45°  with  the  ground 
line,  and  locate  it  so  that  it  impinges  on 
the  surface  of  the  cylinder  at  the  extreme 
visible  element  to  the  left.  The  corre- 
sponding horizontal  projection  of  a7  is  a, 
and  the  point  where  this  ray  impinges  is 
a,"  one  point  of  the  required  shadow. 
Consider  point  b,  vertically  projected 
at  b'.  The  projections  of  its  ray  will  in- 
tersect the  surface  of  the  cylinder  at  the 
point  b"  along  an  element  shown  ver- 
tically projected  at  b'".  Select  another 
point  c'  in  the  vertical  projection,  so  that 
the  ray  c'd'  will  be  tangent  to  the  cyl- 
inder. The  element  is  horizontally  pro- 
jected at  c"d  and  the  ray  impinges  at  c" 
which  is  still  another  point  of  the  shadow. 

In  practice,  several  more  points  are  determined,  but  they  are 
omitted  here  for  the  sake  of  clearness.  The  portion  of  the 
cylinder  away  from  the  source  of  light  must  be  in  the  shade 
and  d'  shows  the  vertical  projection  of  one  element  from  which 
the  illuminated  portion  is  separated  from  the  shade.  Accord- 
ingly, the  portion  of  the  cylinder  to  the  right  of  c"d  is  shaded 
entirely.  The  same  effect  occurs  on  the  superimposed  cap,  and, 
hence,  from  the  element  e,  to  the  right,  the  entire  remaining  area 
is  shaded,  being  unilluminated  from  rays  having  projections  which 
make  an  angle  45°  with  the  ground  line. 

1418.  High=light.     When  a  body,  such  as  a  polished  sphere, 
is  subjected  to  a  source  of  light,  one  spot  on  the  sphere  will  appear 


FIG.  262. 


276  PICTORIAL  EFFECTS  OF  ILLUMINATION 

much  brighter  than  the  rest  of  the  sphere.  This  spot  is  called 
the  brilliant  point,  or,  more  commonly,  the  high-light.  The 
effect  is  that  due  to  the  light  being  immediately  reflected  to  the 
eye  with  practically  undiminished  intensity. 

1419.  Incident  and  reflected  rays.     It    is   a  principle    in 
optics  *  that  the  incident  ray  and  the  reflected  ray  make  equal 
angles  with  the  normal  to  the  surface  on  which  the  light  impinges. 
The  tangent  plane  at  the  point  where  the  light  impinges  must  be 
perpendicular  to  the  normal;    the  incident  ray  and  the  reflected 
ray  also  lie  in  the  same  plane  with  the  normal   and    therefore 
this  plane    is  perpendicular   to  the    tangent  plane.     It  may  be 
observed    that    if    a  line  be  drawn  through  any  point  in  space 
parallel  to  the    ray  of    light  or    incident  ray,  and   through  the 
same  point,  another  line  be  drawn  parallel  to  the  reflected  ray, 
then  the  included  angle  will  be  the  same  irrespective  of  the  loca- 
tion of  the  arbitrary  point  chosen.      The  normal  through  this 
point  would  therefore  be  parallel  to  a  normal  drawn  through  any 
other  point  in  space,  and  a  plane  perpendicular  to  one  normal 
would  be  perpendicular  to  all.     By  choosing  the    perpendicular 
plane  so  that  it  touches  the  surface  a  tangent  plane  to  the  sur- 
face is    obtained;  the  point  of  contact  is  then  the   high-light. 
The  foregoing  can  best  be  illustrated  by  an  example. 

1420.  Problem  5.    To  find  the  high-light  on  a  sphere. 

To  use  a  simple  illustration,  the  high-light  on  the  vertical 

projection  only  will  be  deter- 
mined. Fig.  263  shows  this  con- 
struction. Assume  that  the  eye 
is  directed  perpendicularly  to  the 
vertical  plane.  The  light  comes 
in  a  direction  whose  projections 
make  an  angle  of  45°  with  the 
ground  line.  For  convenience, 
take  a  ray  through  the  centre 
of  the  sphere  o';  the  incident 
ray  is  therefore  m'o'.  The  line 
FlG  263  through  the  point  of  sight  is  per- 

pendicular  to  the   plane   of   the 

paper,  and  is  therefore  projected  at  o'.     Revolve  the  plane  of 
*  See  a  text-book  on  Physics. 


IN   OKTHOGRAFHIC  PROJECTION  277 

these  incident  and  reflected  rays  about  mV  until  it  coin- 
cides or  is  parallel  with  the  vertical  plane.  The  perpendic- 
ular to  the  plane  of  the  paper  through  o'  will  fall  to  c" 
and  the  revovled  angle  will  be  c"o'm";  the  angle  m'o'm" 
being  numerically  *  equal  to  35°  16'  for  rays  whose  projections 
make  an  angle  of  45°  with  the  ground  line.  The  point  m"  is 
graphically  located  as  follows:  Assume  that  a  horizontal  plane 
is  passed  through  o',  the  centre  of  the  sphere.  The  distance 
of  the  point  m'  from  this  new  horizontal  plane  is  m'q'  and  this 
distance  does  not  change  in  the  revolution.  Hence,  the  point  M 
in  space  falls  at  m",  om"  being  the  revolved  position  of  the  actual 
ray  of  light. 

The  bisector  of  the  angle  m"o'c"  is  the  normal  to  the  surface 
and  a  plane  tangent  to  the  revolved  position  of  the  sphere  will 
be  tangent  at  the  point  p".  It  is  unnecessary  to  draw  the  plane 
in  this  case  as  the  normal  readily  determines  p".  The  counter 
revolved  position  of  p"  is  p' ;  therefore,  p'  is  the  required  high- 
light. 

1421.  Multiple  highlights.     Before  leaving  this  subject  it 
must  be  noted  that  with  several  sources  of  light  on  an  object, 
there  may  be  several  high-lights,  although  it  is  usual  to  assume 
only  one  on  a  surface  like  the  sphere.     A  corrugated  surface  will 
have  several  high-lights  for  any  one  position  of  the  eye  and  a 
single  source  of  light. 

1422.  Highlights  of  cylindrical  or  conical  surfaces.     A 

cylinder  or  a  cone  can  have  only  one  high-light  for  any  one  posi- 
tion and  direction  of  vision  of  the  eye.  If  the  surface  is  highly 
polished,  it  will  be  the  only  point  visible;  the  object  itself  cannot 
be  distinguished  as  to  the  details  of  its  form.  As  the  eye  is 
directed  anywhere  along  the  surface,  a  new  high-light  is  observed 
for  each  direction  of  vision  and  the  locus  of  these  points  forms 
approximately  a  straight  line. 

1423.  Aerial  effect  of  illumination.     The  foregoing  princi- 
ples are  true  for  the  conditions  assumed.    In  nature,  however,  it  is 
impossible  to  receive  light  from  only  one  source  in  directly  parallel 
rays,  to  the  total  exclusion  of  any  other  light.     Modifications 
must  be  introduced  because  the  surfaces  are  not  optically  true 

*  Computed  by  trigonometry. 


278  PICTORIAL  EFFECTS  OF  ILLUMINATION 

in  the  first  instance.  This  means  that  a  cylinder,  although  made 
round  as  carefully  as  possible,  very  slight  deviations  hardly 
discernible  by  measuring  instruments  are  readily  detected  by 
their  appearance  in  the  light.  The  reflected  light  augments 
the  imperfections,  so  that  to  the  experienced  eye  the  effect  is 
noticeable.  Other  equally  important  facts  are  those  due  to  the 
reflection  from  the  walls  of  the  room,  and  from  other  objects; 
the  diffusion  caused  by  the  light  being  transmitted  through  the 
window-pane,  and  that  reflected  from  dust  particles  suspended 
in  the  air.  All  these  disturbing  influences  tend  to  illuminate 
certain  other  portions  of  the  object  and  actually  do  so  to  such 
an  extent  that  cognizance  of  this  aerial  effect  of  illumination 
must  be  taken. 

1424.  Graduation  of  shade.     The  effect  on  a  highly  polished 
sphere  can  be  brought  out  by  holding  one  in  a  room  whose  walls 
are  a  dull  black,  similar  to  a  room  used  for  photometric  testing. 
Even   this  photometer  room  does  not  fulfil  the  requirements  to 
the  last  degree,  as  there  is  no  surface  which  does  not  reflect  at 
least  some  light.     Observing  this  sphere  by  aid  of  a  ray  of  light 
coming  through  any  opening  in  the  room,  the  high-light  will 
alone  be  visible.     If  there  were  no  variation  in  the  amount  of 
light  sent  to  the  eye  from  the  various  points  on  the  object,  due 
to  the  aerial  effect  of  illumination,  it  would  be  impossible  to 
recognize  its  form.     Hence  no  light  can  come  to  the  eye  except 
from  the  high-light  unless  the  aerial  effect  is  present. 

With  these  difficulties  to  contend  with,  it  may  seem  that  the 
foregoing  principles  become  invalid.  Such  is  not  the  case, 
however,  as  the  principles  are  true,  in  the  main,  but  the  proper 
correction  is  made  for  diffused  light  and  a  graduation  in  shade 
is  introduced  so  as  make  the  observer  conscious  of  the  desired 
idea  to  be  conveyed. 

1425.  Shading  rules.    At  times  it  may  even  be  necessary  to 
strain  a  point  in  order  to  bring  out  some  detail  of  the  object 
drawn.     This  latter  is  a  characteristic  of  the  skill  of  the  artist. 
The  rules  for  the  representation  of  shades  are  here  inserted,  being 
taken  from  Mahan's  Industrial  Drawing: 

"  1.  Flat  tints  should  be  given  to  plane  surfaces,  when  in  the 
light,  and  parallel  to  the  vertical  plane;  those  nearest  the  eye 
being  lightest. 


FIG   264. 


FIG.  265. 


FIG.  266.  FIG.  267. 

Examples  of  Graduated  Shades  in  Pictorial  Effects  of  Illumination. 

[To  face  page  278 


IN  ORTHOGRAPHIC  PROJECTION  279 

"  2.  Flat  tints  should  be  given  plane  surfaces,  when  in  the 
shade,  and  parallel  to  the  vertical  plane;  those  nearest  the  eye 
being  darkest. 

"  3.  Graduated  tints  should  be  given  to  plane  surfaces,  when 
in  the  light  and  inclined  to  the  vertical  plane;  increasing  the  shade 
as  the  surfaces  recede  from  the  eye;  when  two  such  surfaces 
incline  unequally  the  one  on  which  the  light  falls  most  directly 
should  be  lightest. 

"  4.  Graduated  tints  should  be  given  to  plane  surfaces,  when 
in  the  shade,  and  inclined  to  the  vertical  plane;  decreasing  the 
shade  as  the  surfaces  recede  from  the  eye." 

1426.  Examples  of  graduated  shades.  In  Figs.  264,  265,  266 
and  267  are  shown  a  prism,  cylinder,  cone  and  sphere  shaded  in 
accordance  with  the  above  rules.  It  will  be  seen  that  no  mistake 
can  occur  as  to  the  nature  of  the  objects,  thus  certifying  to  the 
advisability  of  their  adoption. 

QUESTIONS  ON  CHAPTER  XIV 

1.  What  is  the  purpose  of  introducing  the  pictorial  effects  of  illumina- 

tion? 

2.  What  is  meant  by  line  shading? 

3.  How  are  the  straight  lines  of  an  object  line  shaded?    Give  example. 

4.  How  are  the  curved  lines  of  an  object  line  shaded?    Give  example. 

5.  How  are  the  "sections"  line  shaded?    Give  example. 

6.  How  are  convex  surfaces  line  shaded?  Give  example. 

7.  How  are  concave  surfaces  line  shaded?    Give  example. 

8.  How  are  plane  surfaces  line  shaded?    Give  example. 

9.  How  are  objects  made  evident  to  us? 

10.  What  is  the  source  of  light? 

11.  What  are  light  rays? 

12.  What  conventional  direction  of  ray  is  adopted  in  shading  drawings? 

13.  What  is  the  shade  on  an  object? 

14.  What  is  the  shadow  of  an  object? 

15.  What  is  the  umbra?     Give  example. 

16.  What  is  the  penumbra?    Give  example. 

17.  Why  is  only  the  umbra  used  on  drawings? 

18.  What  is  the  fundamental  operation  of  finding  the  shadow  of  an 

object? 

19.  Construct  the  shadow  of  a  line  that  is  situated  so  as  to  have  a 

shadow  on  the  horizontal  plane. 

20.  Construct  the  shadow  of  a  line  that  is  situated  so  as  to  have  a 

shadow  on  the  vertical  plane. 


280  PICTORIAL  EFFECTS  OF  ILLUMINATION 

21.  Construct  the  shadow  of  a  line  that  is  situated  so  as  to  have  a 

shadow  on  both  principal  planes. 

22.  What  is  a  high-light?     Explain  fully. 

23.  What  is  the  incident  ray? 

24.  What  is  the  reflected  ray? 

25.  What  angular  relation  is  there  between  the  normal  and  the  incident 

and  reflected  rays? 

26.  Do  the  normal,  the  incident  ray  and  the  reflected  ray  lie  in  one 

plane? 

27.  Find  the  high-light  on  a  sphere. 

28.  How  are  multiple  high-lights  produced? 

29.  What  is  the  aerial  effect  of  illumination? 

30.  What  is  meant  by  graduation  of  shade? 

31.  To  what  is  the  graduation  of  shade  due? 

32.  Shade  a  sphere  with  lead  pencil  so  as  to  show  the  high-light  and 

the  graduation  of  the  shade. 

33.  Shade  a  cylinder  with  lead  pencil  so  as  to  show  the  high-light  and 

the  graduation  of  the  shade. 

34.  Shade  a  cone  with  lead  pencil  so  as  to  show  the  high-light  and  the 

graduation  of  the  shade. 

35.  Shade  an  octagonal  prism  with  lead  pencil  so  as  to  show  the  high- 

light and  the  graduation  of  the  shade. 

36.  Construct  the  horizontal  shadow  of  a  cube  resting  on  the  horizontal 

plane. 

37.  Construct  the  horizontal  shadow  of  a  cube  which  is  some  distance 

above  the  horizontal  plane. 

38.  Construct  the  horizontal  shadow  of  a  triangular  prism  which  rests 

on  the  horizontal  plane. 

39.  Construct  the  horizontal  shadow  of  a  hexagonal  prism  which  rests 

on  the  horizontal  plane. 

40.  Construct  the  horizontal  shadow  of  a  pyramid  which  rests  on  the 

horizontal  plane. 

41.  Construct  the  shade  and  horizontal  shadow  cast  by  a  cylinder  with 

a  superimposed  circular  cap. 

42.  Construct  the  shade  and  horizontal  shadow  cast  by  a  cylinder  with 

a  superimposed  square  cap. 

43.  Construct  the  shade  and  horizontal  shadow  cast  by  a*cylinder  with 

a  superimposed  octagonal  cap. 

44.  Construct  the  shade  and  horizontal  shadow  cast  by  an  octagonal 

prism  with  a  superimposed  circular  cap. 

45.  Construct  the  shade  and  horizontal  shadow  cast  by  an  octagonal 

prism  with  a  superimposed  square  cap. 

46.  Construct  the  shade  and  horizontal  shadow  cast  by  an  octagonal 

prism  with  a  superimposed  octagonal  cap. 

47.  Construct  the  shade  and  horizontal  shadow  cast  by  a  cone  which 

rests  on  a  square  base. 

48.  Construct  the  shade  and  horizontal  shadow  cast  by  an  octagonal 

pyramid  which  rests  on  a  square  base. 


IN  ORTHOGRAPHIC  PROJECTION  281 

49.  Construct  the  shadow  cast  by  a  cone  which  is  situated  so  as  to 

cast  a  shadow  on  both  principal  planes. 

50.  Construct  the  shadow  cast  by  an  octagonal  pyramid  which  is  situated 

so  as  to  cast  a  shadow  on  both  principal  planes. 

51.  Construct  the  shadow  cast  by  an  octagonal  prism  which  is  situated 

so  as  to  cast  a  shadow  on  both  principal  planes. 

52.  Construct  the  shadow  cast  by  a  superimposed  cap  on  the  inside 

of  a  hollow  semi-cylinder. 


CHAPTER  XV 


PICTORIAL  EFFECTS  OF  ILLUMINATION  IN  PERSPECTIVE 
PROJECTION 

1501.  Introductory.     The  fundamental  principles  of  the  pic- 
torial effects  of  illumination  are  best  studied  in  orthographic 
projection.     Their  ultimate  use,  however,  is  usually  associated 
with  perspective.     Thus,  in  one  color,  an  attempt  is  made  to 
picture  reality,  whether  it  be  used  for  engineering  purposes,  as 
catalogue  illustrations,  or  whether  it  be  used  for  general  illustrating 
purposes.     The   principles,   established   here,    hold   equally  well 
when  color  is  added  to  the  perspective  (making  it  an  aerial  per- 
spective) and  its  illumination;    but,  as  this  is  recognized  as  a 
distinct  field,  it  will  not  be  considered  in  this  book. 

1502.  Problem  1.    To  draw  the  perspective  of  a  rectangular 
prism  and  its  shadow  on  the  horizontal  plane. 

The  first  step  in  this  case  is  to  draw  the  object  and  its  shadow 
orthographically.     The  object  is  shown  in  the  second  angle  as 

has  been  the  custom  in  perspec- 
tive; the  shadow  is  on  the  hori- 
zontal plane  and  its  construc- 
tion is  carried  out  in  accord- 
ance with  principles  previously 
discussed  (Chap.  XIV).  The 
shaded  area  in  Fig.  268  shows 
the  shadow  so  constructed.  The 
advisability  of  the  180°  revolu- 
tion of  the  horizontal  plane  has 
been  given  (1322),  and  carrying 
this  into  effect  in  the  present 
FIG.  268.  instance,  the  effect  shown  in 

Fig.  269  is  obtained.  The  hori- 
zontal projection  of  the  prism  and  the  shadow  of  the  prism  are 
then  shown  below  the  ground  line.  The  perspective  of  the  prism 

282 


a'd 


dh 


IN  PERSPECTIVE  PROJECTION 


283 


perhaps  be  clear  from  the  illustration  in  Fig.  269  as  all  the 
necessary  construction  lines  have  been  included.  The  steps 
necessary  for  the  construction  of  the  perspective  of  the  shadow, 
however,  will  be  considered. 

The  point  of  sight  is  at  S,  shown  horizontally  projected  at 
s  and  vertically  projected  at  s'.  The  vanishing  points  of  the 
diagonals  are  shown  as  v  and  v';  the  distance  of  v  and  v'  from 
s'  are  equal  to  the  distance  s  above  the  ground  line  (1325).  A 


FIG.  269. 


perpendicular  and  a  diagonal  through  the  point  d"  (the  shadow 
of  D  in  space)  will  intersect  at  d'"  which  is  the  required  perspective 
of  the  shadow  of  this  point.  The  same  procedure  will  locate 
c'"  and  b'".  The  corners  F  and  G  are  the  perspectives  of  those 
points  in  space,  and  as  they  rest  on  the  horizontal  plane,  they 
are  also  the  perspectives  of  their  shadows.  By  joining  Fb"'c'"d'" 
and  H  with  lines  the  complete  outline  of  the  shadow  is  obtained, 
except  in  so  far  as  the  limited  portion  behind  the  prism  from 
H  which  is  hidden  from  the  observer  is  concerned. 


284 


PICTORIAL  EFFECTS  OF  ILLUMINATION 


1503.  General  method  of  finding  the  perspective  of  a 
shadow.*  The  above  method  of  constructing  the  perspectives 
of  shadows  is  perfectly  general,  although  lengthy.  It  is  possible 
to  economize  time,  however,  by  taking  advantage  of  the  method 
of  locating  the  perspective  of  the  shadow  directly.  The  shadow 
of  a  point  in  space  on  any  surface  is  the  piercing  point  of  a  ray 
of  light  through  the  point  on  that  surface;  the  perspective  of  that 
shadow  must  therefore  lie  somewhere  on  the  perspective  of  the 
ray  of  light.  It  will  also  lie  on  the  line  of  intersection  of  the 
plane  receiving  the  shadow  with  a  plane  containing  the  ray 
of  light.  The  perspective  of  this  line  of  intersection  will  also 


FIG.  270. 


contain  the  perspective  of  the  shadow.  Hence,  the  perspective 
of  the  shadow  of  the  point  will  lie  on  the  intersection  of  these 
two  perspectives  (1318). 

1504.  Perspectives  of  parallel  rays  of  light.  The  rays 
of  light  are  assumed  as  coming  in  parallel  lines;  being  parallels, 
they  therefore  have  a  common  vanishing  point.  To  find  this 
vanishing  point  (1313)  draw  through  the  point  of  sight,  a  line 
parallel  to  these  rays;  and  the  piercing  point  of  this  line  on  the 
picture  plane  will  be  the  required  vanishing  point.  In  Fig. 
270,  if  a  ray  be  drawn  through  the  point  of  sight,  then  the  vertical 

*  This  method  is  similar,  in  general,  to  the  finding  of  the  piercing  point 
of  a  given  line  on  a  given  plane.  See  Art.  823. 


IN  PERSPECTIVE  PROJECTION  285 

projection  of  the  ray  will  be  sY;  the  horizontal  projection  of 
the  ray  will  be  sr  (a  careful  note  being  made  of  the  180°  revo- 
lution of  this  plane  and  hence  the  revolved  direction  of  the  ray) 
and  the  piercing  point  on  the  picture  plane  will  therefore  be 
atr7. 

1505.  Perspective  of  the  intersection  of  the  visual  plane 
on  the  plane  receiving  the  shadow.     The  line  of  intersection 
of  the  plane  receiving  the  shadow  and  the  plane  containing  the 
ray  of  light  (or  visual  plane  as  this  latter  plane  is  called)  is  in 
our  case  a  horizontal  line,  as  the  plane  receiving  the  shadow  is 
the  horizontal  plane.     If  the  horizontal  projecting  plane  of  the 
ray  be  taken,  it  is  known  that  the  trace  makes  an  angle  of  45° 
with  the  ground  line,  and  that  this  horizontal  line  must  vanish 
in  the  horizon  at  v.     It  will  be  observed  that  this  same  point 
(v)  is  also  the  vanishing  point  of  all  diagonals  drawn  to  the 
right  of  the  point  of  sight.     A  further  note  may  be  taken  of  the 
fact  that  if  the  perspective  of  the  horizontal  projection  of  the 
point  be  joined  with  the  right  vanishing  point  of  the  diagonal, 
the  perspective  is  identical  with  the  perspective  of  the  intersection 
of  the  horizontal  plane  and  the  visual  plane,  because  the  per- 
spectives of  the  horizontal  projection  of  the  point  and  the  van- 
ishing point  are  common  to  the  two. 

1506.  Application  of  the  general  method  of  finding  the 
perspective  of  a  shadow.     The  foregoing  can  be  applied  to  the 
finding  of  the  shadow  of  Problem  1.     A  reference  to  Fig.  270 
in  addition  to  what  follows,  will  indicate  the  application.     Suppose 
the  perspective  of  the  shadow  of  D  is  under  consideration.     The 
perspective  of  the  visual  ray  is  Dr';   the  perspective  of  the  hori- 
zontal projection  of  the  ray  is  Hv;    their  intersection  is  d'", 
the  perspective  sought.     Likewise,  c'"  is  similarly  located,  and 
it  is  the  perspective  of  the  shadow  of  C,  found  by  the  inter- 
section of  the    Cr'  (perspective  of   the    ray)  and    Gv  (perspec- 
tive of   the    horizontal   projection   of   the   ray).     The  point   B 
has  its  shadow  at  b'",  and  its  location  is  clearly  shown  in  the 
figure. 

The  shadow  is  completed  by  joining  the  proper  points  with 
lines.  In  every  way  it  is  identical  with  the  shadow  determined 
in  Problem  1. 


286 


PICTORIAL  EFFECTS  OF  ILLUMINATION 


1507.  Problem  2.  To  draw  the  perspective  of  an  obelisk 
with  its  shade  and  shadow. 

Let  AFGHK  be  the  obelisk  (Fig.  271)  and  S  the  point  of 
sight.  The  perspective  is  drawn  in  the  usual  way.  The  problem 
is  of  interest  in  so  far  as  the  rays  of  light  make  an  angle  of  30° 
with  the  ground  line,  thus  causing  a  longer  shadow  than  when 
the  45°  ray  is  used. 

Through  the  horizontal  and  vertical  projections  of  the  point 
of  sight,  draw  a  ray  parallel  to  the  conventional  ray  adopted. 
This  ray  pierces  the  picture  plane  at  r'  the  vanishing  point  of  all 
the  rays  in  space.  The  horizontal  projections  of  all  rays  vanish 


lie 


FIG.  271. 

at  r"  on  the  horizon,  as  all  lines  should  that  are  parallel  to  the 
horizontal  plane  and  at  the  same  time  belong  to  the  system 
of  lines  parallel  to  the  rays  of  light. 

Inspection  of  Fig.  271  will  show  that  the  lines  GC,  CA,  AE, 
and  EK  affect  the  shadow,  and  that  the  only  points  to  be  located 
for  the  shadow  are  C,  A,  and  E.  To  locate  c",  the  shadow  of 
point  C,  draw  the  perspective  of  the  ray  through  C;  Cr'  is  this 
perspective.  The  perspective  of  the  horizontal  projection  is 
c'"r";  c'"  is  the  perspective  of  the  horizontal  projection  of  C, 
the  necessary  construction  lines  being  shown  in  the  figure.  The 
intersection  of  these  two  perspectives  is  c",  the  required  per- 
spective of  the  shadow  of  C  in  space. 

The  point  A  has  the  perspective  of  its  shadow  at  a"  which, 


• 


FIG.  273.— Commercial  Application  of  the  Pictorial  Effects  of  Illumination  in  Perspective. 

[To  face  page  287] 


IN  PERSPECTIVE  PROJECTION  287 

as  before,  is  the  intersection  of  the  perspective  Ar'  (of  the  ray) 
and  a'"r"  (of  the  horizontal  projection  of  the  ray).  The  point 
E  has  its  shadow  e"  located  in  an  identical  manner  as  the 
preceding  points  c"  and  a". 

As  the  obelisk  is  resting  on  a  plane  (the  horizontal  in  this 
case)  the  base  is  its  own  shadow,  and  it  is  only  necessary  to  join 
the  shadow  of  C  with  G  and  the  line  of  the  shadow  c"G  is  deter- 
mined. Likewise,  join  c"  with  a",  a"  with  e",  and,  finally, 
e"  with  K. 

1508.  Commercial  application  of  the  pictorial  effects  of 
illumination  in  perspective.  A  few  general  remarks,  in  cases 
where  the  shadow  falls  on  itself  or  nearby  objects,  may  not  be 
amiss.  The  draftsman  usually  has  some  choice  in  the  selection 
of  the  direction  of  the  rays,  and,  sometimes,  in  the  location  of 
nearby  objects. 

Where  the  shadow  is  cast  on  the  object  itself  or  on  neighboring 
objects,  it  will,  in  general,  be  found  much  easier  to  find  the  shadow, 
orthographically,  and  then  to  proceed  with  the  making  of  a 
perspective  from  it.  When  the  shadow  is  cast  on  a  horizontal 
surface  only,  the  general  method  outlined  in  Arts.  1503,  1504, 
and  1505  will  find  ready  application. 

The  application  of  the  pictorial  effects  of  illumination  in 
perspective  in  general,  requires  some  consideration  of  the  time 
required  to  make  the  drawings.  The  principles  developed  serve 
as  a  useful  guide,  so  that  the  draftsman  does  not  picture  impos- 
sible shadows,  even  though  the  correct  outline  is  not  given. 
In  fact,  it  is  a  difficult  matter  exactly  to  determine  the  assumed 
direction  of  the  light  from  the  picture  itself.  For  artistic  reasons, 
certain  portions  of  an  object  are  purposely  subdued  in  order  more 
strongly  to  emphasize  some  particular  feature.  The  largest 
application  of  these  principles  lies  in  making  illustrations  for 
high-class  catalogues,  particularly  for  machinery  catalogues.  Figs. 
272  and  273  give  examples  of  this  kind  of  work.  It  will  be  observed 
that  the  presented  principles  are  ignored  in  many  respects,  yet, 
the  effect  is  pleasing  notwithstanding.  After  all,  the  theory 
indicates  correct  modes  of  procedure,  but  the  time  required  to 
make  such  drawings  is  frequently  prohibitive.  Hence,  common 
sense,  based  on  mature  judgment,  must  be  used  as  a  guide. 


288  PICTORIAL  EFFECTS  OF  ILLUMINATION 


QUESTIONS  ON  CHAPTER  XV 

1.  When  rays  of  light  are  parallel,  do  their  perspectives  have  a  common 

vanishing  point  on  the  picture  plane?    Why? 

2.  How  is  the  vanishing  point  of  the  perspectives  of  a  parallel  system 

of  lines  found? 

3.  What  is  a  visual  plane? 

4.  Show  that  the  perspective  of  the  intersection  of  a  visual  plane  and 

the  horizontal  plane  vanishes  on  the  horizon. 

5.  If  the  conventional  direction  of  rays  is  used,  show  that  the  per- 

spectives of  their  horizontal  projections  vanish  in  the  right  diagonal 
vanishing  point. 

6.  State  the  general  method  of  finding  the  perspective  of  a  shadow 

without  first  constructing  it  orthographically. 

7.  Show  that  the  method  of  Question  6  is  an  application  of  finding 

the  perspectives  of  intersecting  lines. 

8.  A  rectangular  prism  rests  on  the  horizontal  plane.     Find  its  shadow 

on  that  plane  by  constructing  it  orthographically  and  then  make 
a  perspective  of  it. 

9.  Take  the  same  prism  of  Question  8  and  construct  its  horizontal 

shadow  directly. 

10.  An  obelisk  rests  on  the  horizontal  plane.     Find  its  shadow  on  that 

plane  by  constructing  'it  orthographically  and  then  make  a  per- 
spective of  it. 

11.  Take  the  same  obelisk  of  Question  10  and  construct  its  horizontal 

shadow  directly. 

Note.    In  the  following  problems  construct  the  shade  and  shadow 
orthographically  and  then  find  its  perspective. 

12.  A  square-based  pyramid  rests  on  a  square  base.      Construct  the 

shadow. 

13.  A  square-based  pyramid  rests  on  a  circular  base.      Construct  the 

shadow. 

14.  A  square-based  pyramid  rests  on  a  hexagonal  base.     Construct  the 

shadow. 

15.  A  cone  rests  on  a  hexagonal  base.     Construct  the  shadow. 

16.  A  rectangular  prism  rests  on  two  square  bases  (stepped).     Construct 

the  shadow. 

17.  A  rectangular  prism  rests  on  two  circular  bases  (stepped).     Construct 

the  shadow. 

18.  A  rectangular  prism  rests  on  two  hexagonal  bases  (stepped).     Con- 

struct the  shadow. 

19.  A  cylinder  has  a  superimposed  square  cap.     Construct  the  shadow. 

20.  A  cylinder  has  a  superimposed  circular  cap.     Construct  the  shadow. 

21.  A  cylinder  has  a  superimposed  hexagonal  cap.    Construct  the  shadow. 

22.  A  cylinder  has  a  superimposed  square  cap  and  rests  on  a  square 

base.     Construct  the  shadow. 


IN  PERSPECTIVE  PROJECTION  289 

23.  A  cylinder  has  a  superimposed  circular  cap  and  rests  on  a  circular 

base.    Construct  the  shadow. 

24.  A  cylinder  has  a  superimposed  hexagonal  cap  and  rests  on  a  hexagonal 

base.     Construct  the  shadow. 

25.  A  hollow  semi-cylinder  has  a  superimposed  cap.     Construct  the 

shadows  on  the  inside  of  the  cylinder  and  on  the  horizontal  plane. 


INDEX 


A  ART.   NO 

Aerial  effect  of  illumination 1423 

Aerial  perspective 1307 

Altitude  of  a  cone " 1004 

Altitude  of  a  cylinder 1009 

Angle  between  curves • 921 

Angle  between  a  given  line  and  a  given  plane 831 

Angle  between  a  given  plane  and  a  principal  plane 828 

Angle  between  two  given  intersecting  lines 826 

Angle  between  two  given  intersecting  planes 827 

Angle,  Line  drawn  through  a  given  point  and  lying  in  a  given  plane 

intersecting  the  first  at  a  given 836 

Angle  of  convergence  of  building  lines  (perspective) 1331 

Angle  of  orthographic  projection.     Advantage  of  the  third 315 

Angle  of  orthographic  projection.     First 315 

Angle,  Through  a  given  line  in  a  given  plane  pass  another  plane  intersect- 
ing it  at  a  given 837 

Angle,  Visual 1304 

Angles  formed  by  the  principal  planes 502 

Angles,  Isometric  projection  of 406 

Angles,  Lines  in  all 514 

Angles,  Oblique  projection  of 207 

Angles  of  orthographic  projection 315 

Angles,  Points  in  aU 516 

Angles  projection  to  drawing.     Application  of 317 

Angles,  Traces  of  planes  in  all 609 

Angles  with  the  planes  of  projection.    Through  a  given  point  to  draw  a 

line  of  a  given  length  and  making  given 834 

Angles  with  the  principal  planes.     Through  a  given  point  draw  a  plane 

making  given 835 

Application  of  angles  of  projection  to  drawing  .  .  . 317 

Application  of  axonometric  projection.     Commercial 412 

Application  of  drawing.     Commercial 104 

Application  of  oblique  projection.     Commercial 212 

Application  orthographic  projection.     Commercial 318 

Application  perspective  projective.     Commercial 1331 

Approximate  method  of  drawing  an  ellipse 906 

Archimedian  spiral 912 

291 


292  INDEX 

ART.  NO. 

Arch,  Perspective  of 1329 

Arc  of  a  circle 905 

Art  of  drawing 102 

Asymptote  to  a  hyperbola 908 

Asymptotic  surface 1030 

Axes,  Angular  relation  between  dimetric 409 

Axes,  Angular  relation  between  isometric 402 

Axes,  Angular  relation  between  trimetric 410 

Axes,  Dimetric 409 

Axes,  Direction  of  dimetric 409 

Axes,  Direction  of  i&ometric 404 

Axes,  Direction  of  trimetric 410 

Axes,  Isometric 402 

Axes,  Trimetric 410 

Axis,  Conjugate,  of  a  hyperbola 908 

Axis,  Major,  of  an  ellipse 906 

Axis,  Minor,  of  an  ellipse 906 

Axis  of  a  cone 1004 

Axis  of  a  cylinder 1009 

Axis  of  a  helix 915 

Axis  of  a  parabola 907 

Axis,  Principal,  of  a  hyperbola  (footnote) 908 

Axis,  Surfaces  of  revolution  having  a  common 1025 

Axis,  Transverse,  of  a  hyperbola 908 

Axonometric  drawing.     Commercial  application  of 412 

Axonometric  drawing  defined 411 

Axonometric  projection.     Commercial  application  of 412 

Axonometric  projection  defined 411 

B 

Base  of  a  cone 1004 

Base  of  a  cylinder 1009 

Bell-surface.     Intersection  of  with  a  plane 1116 

Bounding  figures  in  isometric  projection 408 

Bounding  figures  in  oblique  projection 210 

Bounding  figures  in  perspective  projection 1331 

Branches  of  a  hyperbola 908 

Building,  Perspective  of 1330 

C 

Centre  of  a  circle 905 

Centre  of  a  curvature 924 

Chord  of  a  circle 905 

Circle  defined 905 

Circle,  Graduation  of  the  isometric 407 

Circle,  Involute  of 930 


INDEX  293 

AKT.NO. 

Circle,  Isometric  projection  of 405 

Circle,  Oblique  projection  of 206 

Circle,  Osculating 924 

Circle,  Projection  of,  when  it  lie&  in  an  oblique  plane  the  diameter  and 

centre  of  which  are  known 838 

Circular  cone 1004 

Circular  cylinder 1009 

Classification  of  lines 916 

Classification  of  projections 413,  1332 

Classification  of  surfaces 1031 

Coincident  projections,  Lines  with 515 

Coincident  projections,  Points  with 517 

Commercial  application  of  axonometric  projection 412 

Commercial  application  of  drawing 104 

Commercial  application  of  oblique  projection 212 

Commercial  application  of  orthographic  projection 318 

Commercial  application  of  perspective  projection 1331 

Commercial  application  of  pictorial  effects  of  illumination 1508 

Concave  surfaces,  Doubly 1022 

Concave  surfaces,  Line  shading  applied  to 1406 

Concavo-convex  surface 1022 

Concentric  circles 905 

Concepts,  Lines  and  points  considered  as  mathematical 901,  1001 

Concepts,  Mathematical 501,  1001 

Cone,  Altitude  of 1004 

Cone  and  sphere,  Intersection  of 1220 

Cone,  Axis  of 1004 

Cone,  Base  of 1104 

Cone,  Circular 1004 

Cone  defined 1004 

Cone,  Development  of  an  intersecting  cylinder  and 1219 

Cone,  Development  of  an  oblique 1119 

Cone,  Development  when  intersected  by  a  plane 1113 

Cone,  High-light  on 1422 

Cone,  Intersection  of  a  cylinder  and 1216,  1217,  1218 

Cone,  Intersection  of  a  right  circular,  and  a  plane 1112 

Cone,  Intersection  of  a  sphere  and 1220 

Cone  of  revolution 1004 

Cone,  Representation  of 1005 

Cone,  Right 1004 

Cone,  Right  circular 1004 

Cone,  Slant  height  of 1004 

Cone,  To  assume  an  element  on  the  surface 1006 

Cone,  To  assume  a  point  on  the  surface 1007 

Conical  helix 915 

Conical  surfaces 1004 

Conical  surfaces,  Application  of 1113 


•294  INDEX 

ART.  NO. 

Conical  surfaces,  Line  of  intersection  of 1211,  1212,  1213,  1214 

Conical  surfaces,  Types  of  line  of  intersection  of 1215 

Conjugate  axis  of  a  hyperbola 908 

Contact  of  tangents,  Order  of 923 

Convergence  of  building  lines.     Angle  of 1331 

Convergent  projecting  lines 1302 

Convex  surface.     Doubly  1022 

Convex  surface.     Line  shading  applied  to  1405 

Convolute  surface 1013 

Craticulation 1331 

Cube,  Perspective  of 1317t  1326 

Cube,  Shadow  of 1414 

Curvature,  Centre  of 924 

Curvature,  Radius  of 924 

Curve,  Direction  of 920 

Curve,Plane 903 

Curved  lines,  Doubly : 913 

Curved  lines,  Line  shading  applied  to 1403 

Curved  lines,  Representation  of  doubly 914 

Curved  lines,  Representation  of  singly 904 

Curved  lines,  Singly 903 

Curved  surfaces,  Development  of  doubly 1124 

Curved  surfaces,  Development  of  doubly  (gore  method) 1125,  1127 

Curved  surfaces,  Development  of  doubly  (zone  method) 1125 

Curved  surfaces,  Doubly 1020 

Curved  surfaces,  Intersection  of  doubly,  with  planes 1115 

Curved  surfaces  of  revolution,  Doubly 1022 

Curved  surfaces  of  revolution,  Intersection  of  two  doubly 1222 

Curved  surfaces  of  revolution,  Representation  of  doubly 1026 

Curved  surfaces  of  revolution,  Singly 1021 

Curved  surfaces  of  revolution,  To  assume  a  point  on  a  doubly 1027 

Curved  surfaces,  Singly 1019 

Curves,  Angle  between 921 

Curves,  Smooth 921 

Cycloid,  denned '. 909 

Cylinder,  defined 1009 

Cylinder,  Development  of  an  intersecting  cone  and 1219 

Cylinder,  Development  of  an  oblique 1120 

Cylinder,  Development  when  intersected  by  a  plane 1108 

Cylinder,  High -light  on 1422 

Cylinder,  Intersection  of  a  cone  and 1216,  1217,  1218 

Cylinder,  Intersection  of  a  right  circular,  and  a  plane 1107 

Cylinder,  Intersection  of  a,  with  a  sphere 1221 

Cylinder,  Representation  of 1010 

Cylinder,  Shade  and  shadow  on 1417 

Cylinder,  To  assume  an  element  on  the  surface 1011 

Cylinder,  To  assume  a  point  on  the  surface 1012 


I  INDEX  295 

ART.  NO. 

Cylinders,  Development  of  two  intersecting 1205,  1207,  1210 

Cylinders,  Intersection  of  two 1204,  1206,  1209 

Cylindrical  helix 915 

Cylindrical  surfaces 1008 

Cylindrical  surfaces,  Application  of 1109,  1208 


D 

Developable  surface 1028,  1104 

Development  by  triangulation 1117 

Development  of  a  doubly  curved  surface  by  approximation 1124 

Development  of  a  doubly  curved  surface  by  the  gore  method 1127 

Development  of  a  cone  intersected  by  a  plane 1113 

Development  of  a  cylinder  intersected  by  a  plane 1108 

Development  of  an  intersecting  cone  and  cylinder 1219 

Development  of  an  oblique  cone 1119 

Development  of  an  oblique  cylinder 1120 

Development  of  an  oblique  pyramid 1118 

Development  of  a  prism  intersected  by  a  plane 1106 

Development  of  a  pyramid  intersected  by  a  plane .  . 1112 

Development  of  a  sphere  by  the  gore  method 1121 

Development  of  a  sphere  by  the  zone  method 1 126 

Development  of  a  transition  piece  connecting  a  circle  with  a  square.. . .  1125 

Development  of  a  transition  piece  connecting  two  ellipses. 1123 

Development  of  surfaces 1103 

Development  of  two  intersecting  cylinders 1205,  1207,  1210 

Development  of  two  intersecting  prisms 1203 

Diagonal  vanishing  points.     Location  of 1325 

Diagonal  when  applied  to  perspective 1319 

Diagrams,  Transfer  of,  from  orthographic  to  oblique  projection 505,  608 

Diameter  of  a  circle 905 

Dimensions  on  an  orthographic  projection 308 

Dimetric  axes,  Angular  relation  between 409 

Dimetric  drawing 409 

Dimetric  projection « 409 

Direction  of  a  curve 920 

Direction  of  light  rays,  Conventional 1409 

Directrix  of  a  cycloid 909 

Directrix  of   an  epycycloid 910 

Directrix  of  an  hypocycloid 911 

Directrix  of  a  parabola 907 

Directrix  of  surface 1002,  1003,  1008 

Directrix,  Rectilinear 1003 

Distance  between  a  given  plane  and  a  plane  parrallel  to  it 829 

Distance  between  a  given  point  and  a  given  line 825 

Distance  between  a  given  point  and  a  given  plane 824 

Distance  between  two  points  in  space 819,  820,  821,  822 


296  INDEX 

AKT.   NO. 

Distance  between  two  skew  lines 832 

Distortion  of  oblique  projection 211 

Drawing  an  ellipse,  Approximate  method  of 405 

Drawing  an  ellipse,  Exact  method  of 906 

Drawing,  Application  of  angles  of  projection  to 317 

Drawing,  Application  of  the  physical  principles  of  light  to 1412 

Drawing,  Art  of '. 102 

Drawing,  Axonometric 411 

Drawing,  Commercial  application  of 104 

Drawing,  Commercial  application  of  axonometric 412 

Drawing,  Dimetric 409 

Drawing,  Distinction  between  isometric  projection  and  isometric 403 

Drawing,  Examples  of  isometric 408 

Drawing,  Nature  of 101 

Drawing  of  a  line 502 

Drawing,  Scales  used  in  making 209 

Drawing,  Science  of 102 

Drawing  to  scale 209 

Drawing,  Trimetric 410 

Drawing,  Use  of  bounding  figures  in  isometric 408 

Doubly  concave  surface 1022 

Doubly  convex  surface 1022 

Doubly  curved  line 913 

Doubly  curved  line,  Representation  of 914 

Doubly  curved  surface 1020 

Doubly  curved  surface,  Development  by  approximation 1124 

Doubly  curved  surface,  Development  by  the  gore  method 1127 

Doubly  curved  surfaces  of  revolution 1022 

Doubly  curved  surfaces  of  revolution,  Intersection  of 1222 

Doubly  curved  surfaces  of  revolution,  Intersection  of  by  a  plane 1115 

Doubly  curved  surfaces  of  revolution,  Representation  of 1026 

Doubly  curved  surfaces  of  revolution,  To  assume  a  point  on  a 1027 

E 

Eccentric  circles 905 

Eccentricity  of  circles 905 

Element,  Mathematical ». 501 

Element  of  a  surface 1002 

Element  on  the  surface  of  a  cone,  To  assume 1006 

Element  on  the  surface  of  a  cylinder,  To  assume 1011 

Elevation,  defined 301 

Ellipse,  Approximate  method  of  drawing 405 

Ellipse,  as  isometric  projection  of  a  circle 405 

Ellipse  as  oblique  projection  of  a  circle 206 

Ellipse  denned 906 

Ellipse,  Extract  method  of  drawing 906 


INDEX  297 


Epycycloid,  defined 910 

Evolute,  defined 929 

Examples  of  graduated  shades 1426 

F 

Figures  in  isometric  projection,  Use  of  bounding 408 

Figures  in  oblique  projection,  Use  of  bounding 210 

Figures  in  perspective  projection,  Use  of  bounding 1331 

Foci  of  an  ellipse 906 

Foci  of  an  hyperbola 908 

Focus  of  an  ellipse 906 

Focus  of  an  hyperbola 908 

Focus  of  a  parabola 907 

Frustum  of  a  cone 1004 

G 

Generatrix  of  a  surface 1002,  1003,  1008 

Gore  method,  Development  of  a  doubly  curved  surface 1127 

Gore  method,  Development  of  a  sphere 1125 

Graduation  shades,  Examples  of 1426 

Graduation  in  shade 1424 

Graphic  representation  of  objects 101 

Ground  line,  defined 302,  502 

Ground  line,  Traces  of  a  plane  intersecting  it 606 

Ground  line,  Traces  of  a  plane  parallel  to  it 602 

H 

Helix,  Conical 915 

Helix  defined 915 

Helix,  Uniform  cylindrical 915 

Height  of  a  cone,  Slant 1004 

Helicoid  (footnote) 1014 

Helicoidal  screw  surface,  Oblique 1014 

Helicoidal  screw  surface,  Right 1015 

High-light 1418 

High-light  on  a  cylinder  or  a  cone 1422 

High-light  on  a  sphere 1420 

High-light,  Multiple 1421 

Horizon  defined 1314 

Horizontal  lines  inclined  to  the  picture  plane,  Perspectives  of  parallel  -.  .  . .  1313 

Horizontal  plane,  Revolution  of  in  orthographic  projection 303 

Horizontal  plane,  Revolution  of  in  perspective  projection 1322 

Horizontal  projection  defined 302 

Hyperbola,  defined 908 

Hypocycloid,  defined 911 


298  INDEX 


Illumination,  Aerial  effect  of 1423 

Illumination,  Commercial  application  of  the  pictorial  effect  of 1508 

Incident  rays 1419 

Inclined  lines,  Isometric  projection  of 406 

Inclined  lines,  Oblique  projection  of 207 

Inclined  planes,  Traces  of 605 

Inflexion,  Point  of 926 

Inflexional  tangent 926 

Interpenetration  of  solids 1215 

Intersecting  a  given  line  at  a  given  point,  Line 804 

Intersecting  in  space,  Projection  of  lines 702 

Intersecting  lines,  Angle  between  two  given 826 

Intersecting  lines,  Perspectives  of 1318 

Intersecting  planes,  Angle  between  two  given 827 

Intersecting  the  ground  line,  Traces  of  planes 606 

Intersection  of  a  bell-surface  with  a  plane 1116 

Intersection  of  a  cone  and  cylinder 1216,  1217,  1218 

Intersection  of  a  cone  and  sphere 1 220 

Intersection  of  a  doubly  curved  surface  of  revolution  and  a  plane 1115 

Intersection  of  a  given  line  at  a  given  point  and  at  a  given  angle 836 

Intersection  of  a  prism  and  a  plane 1105 

Intersection  of  a  pyramid  and  a  plane 1110 

Intersection  of  a  right  circular  cone  and  a  plane 1112 

Intersection  of  a  right  circular  cylinder  and  a  plane 1 107 

Intersection  of  a  sphere  and  a  cylinder 1221 

Intersection  of  conical  surfaces 1211,  1212,  1213,  1214 

Intersection  of  conical  surfaces.  Types  of 1215 

Intersection  of  lines 922 

Intersection  of  two  cylinders 1204,  1206,  1209 

Intersection  of  two  planes  oblique  to  each  other  and  to  the  principal 

planes 809,  810 

Intersection  of  two  prisms 1202 

Invisible  lines,  Representation  of '. 208 

Involute,  denned 929 

Involute  of  a  circle 930 

Isometric  axes 402 

Isometric  axes,  Angular  relation 402 

Isometric  axes,  Direction  of 404 

Isometric  circle,  graduation  of 407 

Isometric  drawing.     Distinction  between  isometric  projection  and 403 

Isometric  drawing,  Examples  of 408 

Isometric  drawing,  Use  of  bounding  figures  in 408 

Isometric  projection  and  isometric  drawing.     Distinction  between 403 

Isometric  projection  considered  as  a  special  case  of  orthographic 402 

Isometric  projection.    Examples  of 408 


INDEX  299 

ART.  NO. 

Isometric  projection.     Nature  of 401 

Isometric  axes  of  angles 406 

Isometric  projection  circles 405 

Isometric  projection  inclined  lines 406 

Isohietric  projection,  Theory  of 402 

Isometric  projection.     Use  of  bounding  figures 408 

L 

Light,  Application  of  the  physical  principles  of,  to  drawing 1412 

Light,  High ' 1418 

Light,  High  on  a  sphere 1420 

Light,  Multiple  high 1421 

Light,  Perspective  of  parallel  rays  of 1504 

Light,  Physiological  effect  of 1408 

Light  rays,  Conventional  direction  of 1409 

Line,  Angle  between  a  given,  and  a  given  plane 831 

Line,  Convergent  projecting 1302 

Line,  Distance  between  a  given  point  and  a  given 825 

Line,  Doubly  curved 913 

Line,  Drawing  of 502 

Line  fixed  in  space  by  its  projections 503 

Line,  Ground 302,  502 

Line  in  a  plane,  pass  another  plane  making  a  given  angle 837 

Line,  Indefinite  perspective  of 1316 

Line  intersecting  a  given  line  at  a  given  point 804 

Line  intersecting  a  given  line  at  a  given  point  and  given  angle 836 

Line  lying  in  the  planes  of  projection 512 

Line,  Meridian 1024 

Line,  Oblique  plane  through  a  given  oblique 806,  807 

Line  of  a  given  length  making  given  angles  with  planes  of  projection 834 

Line  on  a  given  plane.  Project  a  given 830 

Line,  Orthographic  representation  of 502,  504 

Line  perpendicular  to  a  given  plane  through  a  given  point 814,  815 

Line  perpendicular  to  planes  of  projection 513 

Line,  Perspective  of 1309,  1321,  1324 

Line  piercing  a  given  plane 823 

Line,  Piercing  point  of,  on  the  principal  planes 506 

Line  piercing  the  principal  planes 805 

Line,  Plane  through  a  given  point,  perpendicular  to  a  given 816 

Line,  Plane  through  three  given  points 817 

Line,  Projecting  plane  of 502,  610 

Line,  Representation  of  doubly  curved 914 

Line,  Revolution  of  a  point  about 707,  818 

Line,  Revolution  of  a  skew 1023 

Line  shading  applied  to  concave  surfaces 1406 

Line  shading  applied  to  convex  surfaces 1405 


300  INDEX 

ART.  NO. 

Line  shading  applied  to  curved  lines 1403 

Line  shading  applied  to  plane  surfaces 1407 

Line  shading  applied  to  sections 1404 

Line  shading  applied  to  straight  lines % 1402 

Line,  Singly  curved 903 

Line,  Singly  curved,  Representation  of 904 

Line,  Straight 902 

Lines,  Straight,  Representation  of 904 

Line  through  a  given  point  parallel  to  a  given  line 803 

Line,  Traces  of  a  plane  intersecting  the  ground 606 

Line,  Traces  of  planes  parallel  to  the  ground 602 

Lines,  Angle  between  two  given  intersecting 826 

Lines,  Classification  of 916 

Lines  considered  as  mathematical  concepts 901 

Lines,  Distance  between  two  skew 832 

Lines  in  all  angles,  Projections  of 514 

Lines  in  oblique  planes,  Projection  of 704 

Lines  of  profile  planes,  Projection  of 518 

Lines  intersecting  in  space,  Projection  of 702 

Lines,  Intersection  of 922 

Lines,  Isometric  projection  of  inclined 406 

Lines,  Line  shading  applied  to  curved 1403 

Lines,  Line  shading  applied  to  straight 1402 

Lines,  Non-intersecting  in  space,  Projection  of 703 

Lines,  Oblique  projection  of  inclined 207 

Lines,  Oblique  projection  of  parallel  (footnote) 207 

Lines,  Oblique  projecting 202 

Lines  parallel  in  space,  Projection  of 701 

Lines  parallel  to  both  principal  planes,  Perspectives  of 1311 

Lines  parallel  to  both  principal  planes,  Projection  of ....'. 511 

Lines  parallel  to  the  plane  of  projection,  Oblique  projection  of 202 

Lines  parallel  to  the  principal  planes  and  lying  in  an  oblique  plane ....  705 

Lines  perpendicular  projecting 302 

Lines  perpendicular  to  given  planes.  Projection  of 706 

Lines  perpendicular  to  the  horizontal  plane,  Perspectives  of 1310 

Lines  perpendicular  to  the  plane  of  projection,  Oblique  projections  of.  .  204 

Lines  perpendicular  to  the  vertical  plane,  Perspectives  of   1312 

Lines,  Perspectives  of  intersecting 1318 

Lines,  Perspectives  of  systems  of  parallel 1313 

Lines,  Representation  of  invisible 208 

Lines,  Representation  of  visible 208 

Lines,  Shadows  of 1413 

Lines,  Skew  (footnote) 703 

Lines,  Systems  of,  in  perspective 1313 

Lines  with  coincident  projections 515 

Linear  perspective 1303 

Locus  of  a  generating  point 902 


INDEX  301 

M  ART.   NO. 

Magnitude  of  objects 103 

Mathematical  concepts 501,  901,  1001 

Mathematical  elements 501 

Mechanical  representation  of  the  principal  planes 510 

Meridian  line 1024 

Meridian  plane 1024 

Multiple  high-light 1421 

N 

Nappes  of  a  conical  surface 1003 

Nature  of  drawing 101 

Nature  of  isometric  projection 401 

Nature  of  oblique  projection 231 

Nature  of  orthographic  projection 301 

Nomenclature  of  projections 507 

Non-intersecting  lines  in  space,  Projection  of 703 

Normal  defined 927 

Normal  plane 1018 

O 

Oblique  and  orthographic  projections  compared 309 

Oblique  cone 1004 

Oblique  cylinder 1009 

Oblique  helicoidal  screw  surface 1014 

Oblique  line,  Oblique  plane  through  a  given 806,  807 

Oblique  plane  of  projection 202 

Oblique  plane,  Projection  of  lines  in  an 704 

Oblique  plane,  Projection  of  lines  parallel  to  the  principal    planes    and 

lying  in  an 705 

Oblique  plane  through  a  given  oblique  line 806,  807 

Oblique  plane  through  a  given  point 808 

Oblique  planes,  Intersections  of 809,  810 

Oblique  projection,  Commercial  application  of 212 

Oblique  projection  considered  as  a  shadow 203 

Oblique  projection,  Distortion  of 211 

Oblique  projection,  Examples  of 210 

Oblique  projection,  Location  of  eye  in  constructing 202,  211 

Oblique  projection,  Location  of  object  from  plane  of  projection 202 

Oblique  projection,  Nature  of 201 

Oblique  projection  of  angles 207 

Oblique  projection  of  circles 206 

Oblique  projection  of  inclined  lines 207 

Oblique  projection  of  lines  parallel  to  the  plane  of  projection 202 

Oblique  projection  of  lines  perpendicular  to  the  plane  of  projection ....     204 
Oblique  projection  of  parallel  lines  (footnote) 207 


302  INDEX 

ART.  NO. 

Oblique  projection,  Theory  of 202,  203,  204,  205 

Oblique  projection,  Transfer  of  diagrams  from  orthographic 505,  608 

Oblique  projection,  Use  of  bounding  figures 210 

Oblique  projecting  lines 202 

Objects,  Graphic  representation  of 101 

Objects,  Magnitude  of 103 

Order  of  contact  of  tangents 923 

Orthographic  and  oblique  projections  compared 309 

Orthographic  planes  of  projection 302 

Orthographic  projection,  advantages  of  third  angle 315 

Orthographic  projection,  Angles  of 315 

Orthographic  projection  considered  as  a  shadow 310 

Orthographic  projection,  Commercial  application  of 318 

Orthographic  projection,  Dimension  on 308 

Orthographic  projection,  First  angle  of 315 

Orthographic  projection,  Location  of  eye  in  constructing 304,  316 

Orthographic  projection,  Location  of  object  with  respect  to  the  planes 

of  projections 306 

Orthographic  projection,  Nature  of . .     301 

Orthographic  projection,  Size  of  object  and  its  projection 305 

Orthographic  projection,  Theory  of 302 

Orthographic  projection,  Third  angle  of 315 

Orthographic  projection,  Transfer  of  diagrams  from  oblique 505,  608 

Orthographic  projections,  Simultaneous  interpretation  of 308,  318 

Orthographic  projections,  with  respect  to  each  other,  Location  of 307 

Orthographic  representation  of  lines 502,  504 

Orthographic  representation  of  points 508 

Osculating  circle 924 

Osculating  plane 925 


P 

Pantograph,  Use  of,  in  perspective 1331 

Parabola,  defined 907 

Parallel  lines,  Oblique  projection  of  (footnote) 207 

Parallel  lines,  Orthographic  projection  of 701 

Parallel  lines  to  both  principal  planes,  Perspectives  of 1311 

Parallel  lines  to  planes  of  projection,  Oblique  projection  of 202 

Parallel  lines  to  planes  of  projection,  Orthographic  projection  of 511 

Parallel  plane  at  a  given  distance  from  a  given  plane 829 

Parallel  to  a  given  plane,  Plane  which  contains  a  given  point  and  is.  .  .  813 

Parallel  rays  of  light,  Perspectives  of 1504 

Parallel  systems  of  lines,  Perspectives  of 1313 

Parallel  to  a  given  line,  Line  through  a  given  point 803 

Parallel  to  the  ground  line.     Traces  of  planes 602 

Parallel  to  the  principal  planes.     Traces  of  planes 601 

Penumbra. .                                                                        1411 


INDEX  303 

/ 

ART.   NO. 

Perpendicular  (when  applied  to  perspective) 1319 

Perpendicular  line  through  a  given  point  to  a  given  plane 814,  815 

Perpendicular  lines  to  planes  of  projection,  Projection  of 513 

Perpendicular  lines  to  the  horizontal  plane,  Perspective  of 1310 

Perpendicular  plane  to  a  given  line  through  a  given  point 816 

Perpendicular  projecting  lines 302 

Perpendicular  to  both  principal  planes.     Traces  of  planes 604 

Perpendicular  to  given  planes.     Projection  of  lines 706 

Perpendicular  to  one  of  the  principal  planes.     Traces  of  planes 603 

Perpendicular  to  the  plane  of  projection.     Oblique  projection  of  lines. .     204 

Perspective,  Aerial 1307 

Perspective  and  shadow  of  a  prism 1502 

Perspective,  Commercial  application  of 1331 

Perspective,  Linear 1303 

Perspective  of  a  building 1330 

Perspective  of  a  cube 1317,  1326 

Perspective  of  a  hexagonal  prism 1327 

Perspective  of  a  line 1309,  1321,  1324 

Perspective  of  a  line,  Indefinite 1316 

Perspective  of  a  line  parallel  to  both  principal  planes 1311 

Perspective  of  a  line  perpendicular  to  the  horizontal  plane 1310 

Perspective  of  a  line  perpendicular  to  the  vertical  plane 1312 

Perspective  of  a  point 1315,  1320,  1323 

Perspective  of  a  pyramid 1328 

Perspective  of  a  shadow,  Application  of  the  general  method 1506 

Perspective  of  a  shadow,  General  method  of  finding 1503 

Perspective  of  an  arch 1329 

Perspective  of  an  obelisk  with  its  shade  and  shadow 1507 

Perspective  of  intersecting  lines 1318 

Perspective  of  parallel  rays  of  light 1504 

Perspective  of  parallel  systems  of  lines 1313 

Perspective  of  the  horizontal  intersection  of  the  visual  plane 1505 

Perspective  projection,  Theory  of 1306 

Perspective  sketches 1331 

Picture,  Center  of 1312 

Picture  plane 1303,  1308 

Picture  plane,  Location  of 1308 

Pictures 101 

Pictorial  effects  of  illumination,  Commercial  application  of 1508 

Physiological  effect  of  light 1408 

Plan,  defined .' 301 

Plane,  Angle  between  a  given  line  and  a  given 831 

Plane,  Angle  between  a  given  plane  and  a  principal 828 

Plane  containing  a  circle  of  a  known  diameter    and  center,  Projection 

of 838 

Plane'containing  a  line  intersected  by   another   at   a   given   point   and 

angle 836 


304  INDEX 

ART.  NO. 

Plane,  Corresponding  projection  of  a  given  point  when  in  a  given. . .  811,  812 

Plane  curve 903 

Plane,  Distance  between  a  given  point  and  a  given 824 

Plane  fixed  in  space  by  its  traces 607 

Plane,  Location  of  picture 1308 

Plane  making  given  angles  with  the  planes  of  projection 835 

Plane,  Meridian 1024 

Plane,  Normal 1018 

Plane  of  projection,  Horizontal 302,  502 

Plane  of  projection,  Oblique 202 

Plane  of  projection,  Oblique  projection  of  lines  parallel  to 202 

Plane  of  projection,  Oblique  projection  of  lines  perpendicular  to 204 

Plane  of  projection,  Vertical 302,  502 

Plane,  Osculating 925 

Plane  parallel  to  a  given  plane  at  a  given  distance  from  it 829 

Plane  passed  through  a  line  in  a  plane  making  a  given  angle 837 

Plane,  Perpendicular  line  through  a  given  point  to  a  given 814,  815 

Plane,  Perspective  of  the  horizontal  intersection  of  the  visual 1505 

Plane,  Picture 1303,  1308 

Plane,  Piercing  point  of  a  line  on  the  principal 506,  805 

Plane,  Projecting,  a  given  line  on  a  given 830 

Plane,  Projecting,  of  a  line 502,  610 

Plane,  Revolution  of  the  horizontal  (in  orthographic) 303 

Plane,  Revolution  of  the  horizontal  (in  perspective) 1322 

Plane  surface 1002 

Plane  surfaces,  Line  shading  applied  to 1407 

Plane,  Tangent 1017 

Plane  through  a  given  oblique  line,  Oblique 806,  807 

Plane  through  a  given  point  and  perpendicular  to  a  given  line 816 

Plane  through  a  given  point,  Oblique 808 

Plane  through  three  given  points , 817 

Plane  which  contains  a  given  point  and  is  parallel  to  a  given  plane 813 

Plane,  Visual 1310 

Planes,  Angles  between  two  given  intersecting 827 

Plants,  Angles  formed  by  principal 502 

Planes  in  all  angles,  Traces  of 609 

Planes  inclined  to  both  principal  planes,  Traces  of 605 

Planes  intersecting  the  ground  line,  Traces  of 606 

Planes,  Intersection  of  two  planes  oblique  to  each  other 809,  810 

Planes,  Lines  in  profile 518 

Planes,  Line  piercing  principal 506,  805 

Planes,  Mechanical  representation  of  the  principal 510 

Planes  of  projection,  Line  drawn  through  a  given    point,  length,    and 

angles  with 834 

Planes  of  projection,  Lines  lying  in . 512 

Planes  of  projection,  Location  of  object  with  respect  to 306 

Planes  of  projection,  Orthographic 302,  502 


INDEX  305 

ART.  NO. 

Planes  of  projection,  Parallel  lines  to 511 

Planes  of  projection,  Plane  drawn,  making  given  angles  with 835 

Planes  of  projection,  Principal 302,  502 

Planes  parallel  to  the  principal  planes,  Traces  of  .• 601 

Planes  parallel  to  the  ground  line,  Traces  of 602 

Planes  perpendicular  to  both  principal  planes,  Traces  of 604 

Planes  perpendicular  to  one  of  the  principal  planes,  Traces  of 603 

Planes,  Points  lying  in  the  principal, 509 

Planes,  Principal 502 

Planes,  Profile 311 

Planes,  Projection  of  lines  in  oblique 704 

Planes,  Projection  of  lines  perpendicular  to  given 706 

Planes,  Section 313 

Planes,  Supplementary 314 

Point,  considered  as  mathematical  concept 901 

Point,  Corresponding  projection  when  one  is  given 811,  812 

Point,  Distance  between  it  and  a  given  line 825 

Point,  Distance  between  it  and  a  given  plane 829 

Point,  Distance  between  two,  in  space 819,  820,  821,  822 

Point,  Generating 902 

Point  in  all  angles 516 

Point,  Line  intersecting  another  at  a  given 804 

Point,  Line  perpendicular  to  a  given  plane  through  a  given 814,  815 

Point,  Line  through  a  given,  parallel  to  a  given  line 803 

Point,  Location  of  diagonal  vanishing 1325 

Point  lying  in  the  principal  planes 509 

Point,  Oblique  plane  through  a  given 808 

Point  of  inflexion 926 

Point  of  sight,  Choice  of 1331 

Point  of  tangency,  To  find 919 

Point  on  a  doubly  curved  surface  of  revolution.     To  assume 1027 

Point  on  a  surface  of  a  cone,  To  assume 1007 

Point  on  a  surface  of  a  cylinder,  To  assume 1012 

Point,  Orthographic  representation  of 508 

Point,  Perspective  of 1315,  1320,  1323 

Point,  Piercing,  of  a  given  line  on  a  given  plane 823 

Point,  Piercing  of  a  given  line  on  the  principal  planes 506 

Point,  Plane  through  three  given 817 

Point,  Plane  through  a  given,  perpendicular  to  a  given  line 816 

Point,  Plane  which  contains  a  given,  and  is  parallel  to  a  given  plane .  .  .     813 

Point,  Revolution  of,  about  a  line 707,  818 

Point,  Through  a  given,  draw  a  line  of  given  length  and  angles 834 

Point,  Through  a  given,  draw  a  plane  making  given  angles 835 

Point,  Vanishing 1305 

Point  with  coincident  projections 517 

Principal  axis  of  a  hyperbola  (footnote) 908 

Principal  planes  of  projection 302,  502 


306  INDEX 

ART.   NO. 

Principal  planes,  Angles  formed  by 502 

Principal  planes,  Angle  between  a  given  plane  and 828 

Principal  planes,  Intersection  of  two  planes  oblique  to  each  other  and  to 

the 809,  810 

Principal  planes,  Line  piercing 805 

Principal  planes.  Mechanical  representation  of 510 

Principal  planes,  Piercing  point  of  line  on 506 

Principal  planes,  Points  lying  in 509 

Principal  planes,  Traces  of  planes  inclined  to  both 605 

Principal  planes,  Traces  of  planes  parallel  to 601 

Principal  planes,  Traces  of  planes  perpendicular  to  both 604 

Principal  planes,  Traces  of  planes  perpendicular  to  one  of  the 603 

Prism,  Developments  of  two  intersecting 1203 

Prism,  Developments  when  intersected  by  a  plane: 1106 

Prism,  Intersection  of  two 1202 

Prism,  Intersection  of  with  a  plane 1105 

Prism,  Perspective  of 1327 

Prism,  Perspective  and  shadow  of • .  .  .  1502 

Prism,  Shadow  of 1416 

Profile  planes 311 

Profile  planes,  Lines  in 518 

Profile  projections.  Location  of 312 

Project  a  given  line  on  a  given  plane 830 

Projecting  line,  Convergent 1302 

Projecting  line,  Oblique 202 

Projecting  line,  Parallel 302,  304 

Projecting  plane  of  a  line •„ 502,  610 

Projection,  axonometric 411 

Projection,  axonometric,  Commercial  application  of 412 

Projection,  Classification  of 413,  1332 

Projection,  Dimetric .x 409 

Projection,  Isometric,  and  isometric  drawing  compared 403 

Projection,  Isometric  considered  as  special  case  of  orthographic 402 

Projection,  Isometric,  Examples  of 408 

Projection,  Isometric,  Nature  of 401 

Projection,  Isometric  of  angles 406 

Projection,  Isometric  of  circles 405 

Projection,  Isometric  of  inclined  lines 406 

Projection,  Isometric,  Theory  of 402 

Projection,  Isometric,  Use  of  bounding  figures 408 

Projection,  Oblique,  Commercial  application  of 212 

Projection,  Oblique  considered  as  shadow 203 

Projection,  Oblique,  Distortion  of 211 

Projection,  Oblique,  Examples  of 210 

Projection,  Oblique,  Location  of  object  from  plane 202 

Projection,  Oblique,  Nature  of 201 

Projection,  Oblique  of  angles 207 


INDEX  307 

ART.  NO. 

Projection,  Oblique  of  circles 206 

Projection,  Oblique  of  inclined  lines 207 

Projection,  Oblique  of  lines  parallel  to  plane 202 

Projection,  Oblique  of  lines  perpendicular  to  plane 204 

Projection,  Oblique  of  parallel  lines  in  space  (footnote) 207 

Projection,  Oblique  plane  of 202 

Projection,  Oblique  position  of  eye  in  constructing 202,  211 

Projection,  Oblique,  Theory  of 202,  203,  204,  205 

Projection,  Oblique,  Use  of  bounding  figures 210 

Projection,  Orthographic,  angles  of 315 

Projection,  Orthographic,  Application  of  angles 317 

Projection,  Orthographic,  Commercial  application 318 

Projection,  Orthographic,  Compared  with  oblique 309 

Projection,  Orthographic,  Considered  as  shadow 310 

Projection,  Orthographic,  Dimensions  on 308 

Projection,  Orthographic,  First  angle 315 

Projection,  Orthographic,  Horizontal  plane  of 302,  502 

Projection,  Orthographic,  Line  fixed  in  space  by 503 

Projection,  Orthographic,  Location  of  object  to  planes 306 

Projection,  Orthographic,  Location  of  observer  while  constructing.  .    304,  316 

Projection,  Orthographic,  Location  of  profiles  312 

Projection,  Orthographic,  Location  of  projections  with  respect  to  each 

other 307 

Projection,  Orthographic,  Nature  of 301 

Projection,  Orthographic,  Nomenclature  of 507 

Projection,  Orthographic,  of  a  circle  lying  in  an  oblique  plane 838 

Projection,  Orthographic,  of  a  point  lying  in  a  plane  when  one  projec- 
tion is  given 811,  812 

Projection,  Orthographic  of  lines  in  oblique  planes 704 

Projection,  Orthographic  of  lines  intersecting  in  space 702 

Projection,  Orthographic  of  lines  lying  in  the  planes  of 512 

Projection,  Orthographic  of  lines  non-intersecting  in  space 703 

Projection,  Orthographic  of  lines  parallel  in  space 701 

Projection,  Orthographic  of  lines  parallel  to  planes  of 511 

Projection,  Orthographic  of  lines  parallel  to  one  plane  and    in    an 

oblique  plane , 705 

Projection,  Orthographic  of  lines  perpendicular  to  given  planes 706 

Projection,  Orthographic  of  lines  perpendicular  to  planes  of 513 

Projection,  Orthographic  of  lines  with  coincident 515 

Projection,  Orthographic  of  points  with  coincident 517 

Projection,  Orthographic,  Perpendicular  projecting  lines 302 

Projection,  Orthographic,  Plane  of 302 

Projection,  Orthographic,  Position  of  eye  in  constructing 302,  334,  316 

Projection,  Orthographic,  Principal  planes  of 302 

Projection,  Orthographic,  Size  of  object  and  its  projection 305 

Projection,  Orthographic,  Simultaneous  interpretation 308,  318 

Projection,  Orthographic,  Theory  of 302 


308  INDEX 

AHT.  NO. 

Projection,  Orthographic,  Third  angle  of  projection 315 

Projection,  Orthographic,  Transfer  of  diagrams  from  oblique  to ....   505,  608 

Projection,  Orthographic,  Vertical  plane  of 302,  502 

Projection,  Perspective,  (see  topics  under  perspective) 

Projection,  Scenographic 1302 

Projection,  Theory  of  perspective 1306 

Projection,  Trimetric 410 

Pyramid,  Development  of  an  oblique 1118 

Pyramid,  Development  when  intersected  by  a  plane 1111 

Pyramid,  Intersection  of  with  a  plane 1110 

Pyramid,  Perspective  of 1328 

Pyramid,  Shadow  of 1415 

Q 

Quadrant  of  a  circle 905 

R 

Radius  of  a  circle 905 

Rays,  Conventional  direction  of  light 1409 

Rays,  Incident 1419 

Rays  of  light,  Perspective  of  parallel 1504 

Rays,  Reflected 1419 

Rays,  Visual . 1304 

Rectification,  defined 928 

Reflected  rays 1419 

Representation  of  cones 1005 

Representation  of  cylinders 1010 

Representation  of  doubly  curved  surfaces  of  revolution 1026 

Representation  of  invisible  lines 208 

Representation  of  lines,  Orthographic 502,  504 

Representation  of  objects,  Graphic 101 

Representation  of  points,  Graphic 508 

Representation  of  singly  curved  lines 904 

Representation  of  straight  lines 904 

Representation  of  the  principal  planes,  Mechanical 510 

Representation  of  visible  lines 208 

Revolution,  Doubly  curved  surface  of 1022 

Revolution,  Cone  of 1004 

Revolution,  Cylinder  of 1009 

Revolution,  of  a  point  about  a  line 707,  818 

Revollution  of  a  skew  line 1023 

Revo  ution  of  the  horizontal  plane,  (orthographic) 303 

Revolution  of  the  horizontal  plane,  (perspective) 1322 

Revolution,  Representation  of  doubly  curved  surfaces  of 1026 

Revolution,  Singly  curved  surfaces  of 1021 

Revolution,  Surfaces  of,  having  a  common  axis 1025 


INDEX  309 

ART.  NO. 

Revolution,  To  assume  a  point  on  a  doubly  curved  surface  of 1027 

Right  cone 1004 

Right  cylinder 1009 

Right  helicoidal  screw  surface 1015 

Ruled  surface 1029 

Rules  for  shading 1425 


Scale,  Choice  of 209 

Scale,  Drawing  to 209 

Scales  used  in  making  drawings 209 

Scenographic  projection 1302 

Science  of  drawing 102 

Screw  surface,  Oblique  helicoidal 1014 

Screw  surface,  Right  helicoidal 1015 

Secant  of  a  circle 905 

Section  plane 313 

Sections,  Line  shading  applied  to 1404 

Sector  of  a  circle 905 

Segment  of  a  circle 905 

Semicircle 905 

Shade,  defined 1410 

Shade  and  shadow  of  an  obelisk  in  perspective 1507 

Shade  and  shadow  on  a  cylinder 1417 

Shade,  Graduation  in 1424 

Shades,  Examples  of  graduated 1426 

Shading  applied  to  concave  surfaces,  Line 1406 

Shading  applied  to  convex  surfaces,  Line 1405 

Shading  applied  to  curved  lines,  Line 1403 

Shading  applied  to  plane  surfaces,  Line 1407 

Shading  applied  to  sections,  Line 1404 

Shading  applied  to  straight  lines,  Line 1402 

Shading  rules , 1425 

Shadow  defined 1410 

Shadow,  Application  of  general  method  of  finding  its  perspective ......  1506 

Shadow  and  perspective  of  a  prism 1502 

Shadow  and  shade  of  an  obelisk  in  perspective 1507 

Shadow  and  shade  on  a  cylinder 1417 

Shadow,  General  method  of  finding  its  perspective 1503 

Shadow,  Oblique  projection  considered  as 203 

Shadow  of  a  cube 1414 

Shadow  of  a  line 1413 

Shadow  of  a  prism 1416 

Shadow  of  a  pyramid «. 1415 

Shadow,  Orthographic  projection  considered  as 310 

Si^ht,  Choice  of  point  of 1331 

Singly  curved  line,  defined 903 


310  INDEX 

ART.   NO. 

Singly  curved  line,  Representation  of 904 

Singly  curved  surface  defined 1019 

Singly  curved  surface  of  revolution 1021 

Sketches,  Perspective .' 1331 

Skew  lines,  Distance  between  two 832 

Skew  lines  (footnote) '  703 

Skew  lines,  Revolution  of 1023 

Slant  height  of  a  cone .' 1004 

Solids,  Interpenetration  of 1202,  1215 

Spiral  defined 912 

Spiral,  Archimedian 912 

Sphere  and  cone,  Intersection  of 1220 

Sphere  and  cylinder,  Intersection  of 1221 

Sphere  developed  by  the  gore  method 1125 

Sphere  developed  by  the  zone  method 1126 

Sphere,  High-light  on 1420 

Straight  lines  defined 902 

Straight  lines,  Representation  of 904 

Straight  lines,  Line  shading  applied  to 1402 

Supplementary  plane 314 

Surface,  Asymptotic 1030 

Surface,  Bell,  Intersection  of,  with  a  plane 1116 

Surface,  Concavo-  convex 1022 

Surface,  Conical 1003 

Surface,  Convolute 1013 

Surface,  Cylindrical 1008 

Surface,  Developable 1028,  1103,  1104 

Surface,  Doubly  concave 1022 

Surface,  Doubly  convex  1022 

Surface,  Doubly  curved 1020 

Surface,  Doubly  curved,  Developments  by  approximation 1124 

Surface,  Doubly  curved,  Developments  by  the  gore  method 1127 

Surface,  Doubly  curved,  Intersection  of,  with  a  plane 1115 

Surface,  Oblique  helicoidal  screw 1014 

Surface,  Plane,  defined 1002 

Surface,  Right  helicoidal  screw 1015 

Surface,  Ruled 1029 

Surface,  Singly  curved 1019 

Surface,  Warped 1016 

Surface  of  a  cone,  To  assume  a  point  on  the 1007 

Surface  of  a  cone,  To  assume  an  element  on  the 1006 

Surface  of  a  cylinder,  To  assume  a  point  on  the 1012 

Surface  of  a  cylinder,  To  assume  an  element  on  the 1011 

Surface  of  revolution,  Doubly  curved  / 1022 

Surface  of  revolution,  Singly  curved 1021 

Surface  of  revolution,  To  asusme  a  point  on  a  doubly  curved 1027 

Surfaces,  Application  of  conical 1114,  1211 


INDEX  311 


Surfaces,  Application  of  cylindrical 1109,  1208 

Surfaces,  Classification  of 1031 

Surfaces,  Development  of  doubly  curved 1124 

Surfaces,  Gore  method  of  developing 1124 

Surfaces,  Line  of  intersection  of  conical 1211,  1212,  1213,  1214 

Surfaces,  Line  shading  applied  to  concave 1406 

Surfaces,  Line  shading  applied  to  convex 1405 

Surfaces,  Line  shading  applied  to  plane 1407 

Surfaces  of  revolution  having  a  common  axis 1025 

Surfaces  of  revolution,  Intersection  of  two  doubly 1222 

Surfaces  of  revolution,  Representation  of  doubly  curved 1026 

Surfaces,  Types  of  intersection  of  conical 1215 

Surfaces,  Zone  method  of  developing ' 1124 

System  of  lines  (in  perspective) 1313 


T 

Tangent,  defined 917 

Tangent,  Construction  of ' 918 

Tangent,  Inflexional 926 

Tangent,  Order  of  contact  of 923 

Tangent  plane 1017 

Tangency,  To  find  point  of 919 

Theory  of  isometric  projection 402 

Theory  of  oblique  projection 202,  203,  204,  205 

Theory  of  orthographic  projection 302 

Theory  of  perspective  projection 1306 

Theory  of  shades  and  shadows 1412 

Torus 1022 

Torus,  Development  of 1109 

Traces  of  planes  in  all  angles 609 

Traces  of  planes  inclined  to  both  principal  planes 605 

Traces  of  planes  intersecting  the  ground  line 606 

Traces  of  planes  parallel  to  the  ground  line t 602 

Traces  of  planes  parallel  to  the  principal  planes 601 

Traces  of  planes  perpendicular  to  both  principal  planes 604 

Traces  of  planes  perpendicular  to  one  of  the  principal  planes 603 

Traces,  Plane  fixed  in  space  by 607 

Trammel  method  of  drawing  an  ellipse 906 

Transition  piece,  defined 1121 

Transition  piece  connecting  a  circle  with  a  square,  Development  of.  ...  1122 

Transition  piece  connecting  two  ellipses,  Development  of 1123 

Transverse  axis  of  a  hyperbola 908 

Triangulation,  Development  by 1117 

Trimetric  axes,  angular  relation 410 

Trimetric  drawing 410 

Trimetric  projection 410 


312  INDEX 


ART.   NO. 

Truncated  cone 


Truncated  cylinder 

U 


Umbra,  defined  ................................ 

V 

Vanishing  points,  defined  ................. 

Vanishing  points,  Location  of  diagonal  .............  ]  325 

Vertex  of  a  conical  surface  ...................... 

Vertex  of  a  hyperbola  ........................ 

Vertex  of  a  parabola  ..................... 


Vertical  plane  of  projection  ........................  302  502 

Vertical  projection  ...............................  '  gQr> 

Visible  lines,  Representation  of  ............................  208 

Visual  angle  .............................................  .......   1304 

Visual  plane  .............................  ^Q 

Visual  plane,  Perspective  of  horizontal  intersection  of  the  ............   1505 

Visual  ray  ................  .'  .................................   "  1304 

W 
Warped  surface  ................................................   1016 

Z 

Zone  method  of  developing  a  sphere  ...............................   1125 

Zone  method  of  developing  surfaces  ...........................   1124 


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Addyman,  F.  T.     Practical  X-Ray  Work 8vo,  *4  oo 

Adler,  A.  A.    Principles  of  Projecting-line  Drawing 8vo,  *i  oo 

Theory  of  Engineering  Drawing 8vo,  (In  Press.} 

Aikman,  C.  M.    Manures  and  the  Principles  of  Manuring 8vo,  2  50 

Aitken,  W.     Manual  of  the  Telephone 8vo,  *8  oo 

d'Albe,  E.  E.  F.,   Contemporary  Chemistry i2mo,  *i  25 

Alexander,  J.  H.     Elementary  Electrical  Engineering. . i2mo,  2  oo 

Universal  Dictionary  of  Weights  and  Measures 8vo,  3  50 

"  Alfrec."    Wireless  Telegraph  Designs 

Allan,  W.     Strength  of  Beams  Under  Transverse  Loads.     (Science  Series 

No.  19.) i6mo,  o  50 

Theory  of  Arches.     (Science  Series  No.  1 1.) i6mo, 

Allen,  H.     Modern  Power  Gas  Producer  Practice  and  Applications..  i2mo,  *2  50 

—  Gas  and  Oil  Engines 8vo,  *4  50 

Anderson,  F.  A.     Boiler  Feed  Water 8vo,  *2  50 

Anderson,  Capt.  G.  L.     Handbook  for  the  Use  of  Electricians 8vo,  3  oo 

Anderson,  J.  W.     Prospector's  Handbook i2mo,  i  50 

Ande"s,  L.     Vegetable  Fats  and  Oils 8vo,  *4  oo 

Animal  Fats  and  Oils.     Trans,  by  C.  Salter 8vo,  *4  oo 

Drying  Oils,  Boiled  Oil,  and  Solid  and  Liquid  Driers 8vo,  *5  oo 

Iron  Corrosion,  Anti-fouling  and  Anti- corrosive  Paints.     Trans,  by 

C.  Salter..,                                                                             .  .8vo,  *4  oo 


D.   VAN    NOSTRAND   COMPANY'S    SHORT   TITLE   CATALOG        3 

Ande*s,   L.     Oil  Colors,  and  Printers'   Ink.     Trans,  by  A.   Morris  and 

H.  Robson 8vo,  *2  50 

Treatment   of  Paper  for  Special  Purposes.     Trans,  by  C.  Salter. 

i2mo,  *2  50 

Andrews,  E.  S.    Reinforced  Concrete  Construction i2mo,  *i  25 

Annual  Reports  on  the  Progress  of  Chemistry. 

Vol.     I.  (1904) 8vo,  *2  oo 

Vol.    II.  (IQOS) 8vo,  *2  oo 

Vol.  III.  (1906) 8vo,  *2  oo 

Vol.  IV.  (1907) 8vo,  *2  oo 

Vol.    V.  (1908) 8vo,  *2  oo 

Vol.  VI.  (1909) 8vo,  *2  oo 

Vol.  VII.  (1910) 8vo,  *2  oo 

Argand,   M.     Imaginary   Quantities.     Translated   from   the   French   by 

A.  S.  Hardy.     (Science  Series  No.  52.) i6mo,  o  50 

Armstrong,  R.,  and  Idell,  F.  E.     Chimneys  for  Furnaces  and  Steam  Boilers. 

(Science  Series  No.  i.) i6mo,  o  50 

Arnold,  E.     Armature  Windings  of  Direct-Current  Dynamos.      Trans,  by 

F.  B.  DeGress 8vo,  *2  oo 

Ashe,  S.  W.,  and  Keiley,  J.  D.     Electric  Railways.  Theoretically  and 

Practically  Treated.     Vol.  I.  Rolling  Stock i2mo,  *2  50 

Ashe,  S.  W.     Electric  Railways.     Vol.  II.  Engineering  Preliminaries  and 

Direct  Current  Sub-Stations i2mo,  *2  50 

Electricity:  Experimentally  and  Practically  Applied i2mo,  *2  oo 

Atkinson,  A.  A.     Electrical  and  Magnetic  Calculations 8vo,  *i  50 

Atkinson,  J.  J.     Friction  of  Air  in  Mines.     (Science  Series  No.  14.) . .  i6mo,  o  50 
Atkinson,  J.  J.,  and  Williams,  Jr.,  E.  H.     Gases  Met  with  in  Coal  Mines. 

(Science  Series  No.  13.) i6mo,  o  50 

Atkinson,  P.     The  Elements  of  Electric  Lighting I2mo,  i  50 

The  Elements  of  Dynamic  Electricity  and  Magnetism i2mo,  2  oo 

Power  Transmitted  by  Electricity i2mo,  2  oo 

Auchincloss,  W.  S.     Link  and  Valve  Motions  Simplified 8vo,  *i  50 

Ayrton,  H.     The  Electric  Arc 8vo,  *5  oo 

Bacon,  F.  W.     Treatise  on  the  Richards  Steam-Engine  Indicator  . .  i2mo,  i  oo 

Bailes,  G.  M.     Modern  Mining  Practice.     Five  Volumes 8vo,  each,  3  50 

Bailey,  R.  D.     The  Brewers'  Analyst Svo,  *5  oo 

Baker,  A.  L.     Quaternions Svo,  *i  25 

Thick-Lens  Optics (In  Press., 

Baker,  Ben j.     Pressure  of  Earthwork.     (Science  Series  No.  56.)... i6mo) 

Baker,  I.  0.     Levelling.     (Science  Series  No.  91.) i6mo,  o  50 

Baker,  M.  N.     Potable  Water.     (Science  Series  No.  61.) i6mo,  o  50 

Sewerage  and  Sewage  Purification.     (Science  Series  No.  i8.)..i6mo,  050 

Baker,  T.  T.     Telegraphic  Transmission  of  Photographs i2mc,  *i  25 

Bale,  G.  R.     Modern  Iron  Foundry  Practice.     Two  Volumes.     i2mo. 

Vol.    I.  Foundry  Equipment,  Materials  Used *2  50 

Vol.  II.  Machine  Moulding  and  Moulding  Machines *i  50 

Bale,  M.  P.     Pumps  and  Pumping 1 2mo,  i  50 

Ball,  J.  W.     Concrete  Structures  in  Railways.    (In  Press.) 8vo, 

Ball,  R.  S.     Popular  Guide  to  the  Heavens Svo,  *4  50 

Natural  Sources  of  Power.     (Westminster  Series.) Svo,  *2  oo 


4        D.  VAN  NOSTRAND   COMPANY'S  SHORT  TITLE  CATALOG 

Ball,  W.  V.     Law  Affecting  Engineers 8vo,  *3  50 

Bankson,  Lloyd.     Slide  Valve  Diagrams.     (Science  Series  No.  108.) .  i6mo,  o  50 

Barba,  J.     Use  of  Steel  for  Constructive  Purposes i2mo,  i  oo 

Barham,  G.  B.    Development  of  the  Incandescent  Electric  Lamp.  . . .  (In  Press.) 

Barker,  A.    Textiles  and  Their  Manufacture.    (Westminster  Series.) . .  8vo,  2  oo 

Barker,  A.  H.     Graphic  Methods  of  Engine  Design i2mo,  *i  50 

Barnard,  F.  A.  P.     Report  on  Machinery  and  Processes  of  the  Industrial 

Arts  and  Apparatus  of  the  Exact  Sciences  at  the  Paris  Universal 

Exposition,  1867 8vo,  5  oo 

Barnard,  J.  H.     The  Naval  Militiaman's  Guide i6mo,  leather  i  25 

Barnard,  Major  J.  G.     Rotary  Motion.     (Science  Series  No.  90.) i6mo,  o  50 

Barrus,  G.  H.     Boiler  Tests 8vo,  *3  oo 

Engine  Tests 8vo,  *4  oo 

The  above  two  purchased  together *6  oo 

Barwise,  S.     The  Purification  of  Sewage i2mo,  3  50 

Baterden,  J.  R.     Timber.     (Westminster  Series.) 8vo,  *2  oo 

Bates,  E.  L.,  and  Charlesworth,  F.     Practical  Mathematics i2mo, 

Part  I.    Preliminary  and  Elementary  Course *i  50 

Part  n.    Advanced  Course *i  50 

Beadle,  C.     Chapters  on  Papermaking.     Five  Volumes i2mo,  each,  *2  oo 

Beaumont,  R.     Color  in  Woven  Design 8vo,  *6  oo 

Finishing  of  Textile  Fabrics 8vo,  *4  oo 

Beaumont,  W.  W.     The  Steam-Engine  Indicator 8vo,  2  50 

Bechhold.     Colloids  in  Biology  and  Medicine.    Trans,  by  J.  G.  Bullowa 

(In  Press.) 

Beckwith,  A.     Pottery 8vo,  paper,  o  60 

Bedell,  F.,  and  Pierce,  C.  A.    Direct  and  Alternating  Current  Manual.Svo,  *2  oo 

Beech,  F.     Dyeing  of  Cotton  Fabrics 8vo,  *3  oo 

Dyeing  of  Woolen  Fabrics 8vo,  *3  50 

Begtrup,  J.     The  Slide  Valve ;,;.>  .|. 8vo,  *2  oo 

Beggs,  G.  E.     Stresses  in  Railway  Girders  and  Bridges (In  Press.) 

Bender,  C.  E.     Continuous  Bridges.     (Science  Series  No.  26.) i6mo,  o  50 

Proportions  of  Piers  used  in  Bridges.     (Science  Series  No.  4.) 

i6mo,  o  50 

Bennett,  H.  G.     The  Manufacture  of  Leather 8vo,  *4  50 

Leather  Trades    (Outlines   of  Industrial   Chemistry).    8vo ..  (In  Press.) 

Bernthsen,    A.      A  Text  -  book  of  Organic  Chemistry.      Trans,  by  G. 

M'Gowan i2mo,  *2  50 

Berry,  W.  J.    Differential  Equations  of  the  First  Species.    i2mo.  (In  Preparation.) 
Bersch,  J.     Manufacture  of  Mineral  and  Lake  Pigments.     Trans,  by  A.  C. 

Wright 8vo,  *5  oo 

Bertin,  L.  E.     Marine  Boilers.     Trans,  by  L.  S.  Robertson 8vo,  5  oo 

Beveridge,  J.     Papermaker's  Pocket  Book i2mo,  *4  oo 

Binns,  C.  F.     Ceramic  Technology 8vo,  *5  oo 

Manual  of  Practical  Potting 8vo,  *7  50 

The  Potter's  Craft I2mo,  *2  oo 

Birchmore,  W.  H.    Interpretation  of  Gas  Analysis I2mo,  *i  25 

Elaine,  R.  G.     The  Calculus  and  Its  Applications i2mo,  *i  50 

Blake,  W.  H.     Brewers'  Vade  Mecum 8vo,  *4  oo 

Blake,  W.  P.     Report  upon  the  Precious  Metals 8vo,  2  oo 

Bligh,  W.  G.     The  Practical  Design  of  Irrigation  Works 8vo,  *6  oo 


D.   VAN   NOSTRAND   COMPANY'S   SHORT  TITLE   CATALOG        5 

Bliicher,  H.      Modern  Industrial  Chemistry.     Trans,  by  J.  P.  Millington 

8vo,  *7  50 

Blyth,  A.  W.     Foods:  Their  Composition  and  Analysis 8vo,  7  50 

—  Poisons:  Their  Effects  and  Detection 8vo,  7  50 

Bockmann,  F.     Celluloid i2mo,  *2  50 

Bodmer,  G.  R.     Hydraulic  Motors  and  Turbines i2mo,  5  oo 

Boileau,  J.  T.     Traverse  Tables 8vo,  5  oo 

Bonney,  G.  E.     The  Electro-platers'  Handbook i2mo,  I  20 

Booth,  N.     Guide  to  the  Ring-spinning  Frame i2mo,  *i  25 

Booth,  W.  H.     Water  Softening  and  Treatment 8vo,  *2  50 

—  Superheaters  and  Superheating  and  Their  Control 8vo,  *i  50 

Bottcher,  A.     Cranes:    Their  Construction,  Mechanical  Equipment  and 

Working.     Trans,  by  A.  Tolhausen 4to,  *io  oo 

Bottler,  M.     Modern  Bleaching  Agents.     Trans,  by  C.  Salter i2mo,  *2  50 

Bottone,  S.  R.     Magnetos  for  Automobilists i2mo,  *i  oo 

Boulton,  S.  B.     Preservation  of  Timber.     (Science  Series  No.  82.) .  i6mo,  o  50 

Bourgougnon,  A.     Physical  Problems.     (Science  Series  No.  113.).. i6mo,  050 
Bourry,  E.     Treatise  on  Ceramic  Industries.     Trans,  by  J.  J.  Sudborough. 

8vo,  *5  oo 

Bow,  R.  H.     A  Treatise  on  Bracing 8vo,  i  50 

Bowie,  A.  J.,  Jr.     A  Practical  Treatise  on  Hydraulic  Mining 8vo,  5  oo 

Bowker,  W.  R.     Dynamo,  Motor  and  Switchboard  Circuits 8vo,  *2  50 

Bowles,  O.     Tables  of  Common  Rocks.      (Science  Series  No.  125.). .  i6mo,  o  50 

Bowser,  E.  A.     Elementary  Treatise  on  Analytic  Geometry i2mo,  i  75 

Elementary  Treatise  on  the  Differential  and  Integral  Calculus.  12 mo,  2  25 

Elementary  Treatise  on  Analytic  Mechanics i2mo,  3  oo 

Elementary  Treatise  on  Hydro-mechanics i2mo,  2  50 

A  Treatise  on  Roofs  and  Bridges i2mo,  *2  25 

Boycott,  G.  W.  M.     Compressed  Air  Work  and  Diving 8vo,  *4  oo 

Bragg,  E.  M.     Marine  Engine  Design. i2mo,  *2  oo 

Brainard,  F.  R,     The  Sextant.     (Science  Series  No.  101.) i6mo, 

Brassey's  Naval  Annual  for  191 1 8vo,  *6  oo 

Brew,  W.     Three-Phase  Transmission 8vo,  *2  oo 

Brewer,  R.  W.  A.    Motor  Car  Construction i2mo, 

Briggs,    R.,    and   Wolff,    A.    R.     Steam-Heating.     (Science   Series   No. 

67.) i6mo,  o  50 

Bright,  C.     The  Life  Story  of  Sir  Charles  Tilson  Bright 8vo,  *4  50 

Brislee,  T.  J.    Introduction  to  the  Study  of  Fuel.     (Outlines  of  Indus- 
trial Chemistry) 8vo,  *3  oo 

British  Standard  Sections 8x15  *i  oo 

Complete  list  of  this  series  (45  parts)  sent  on  application. 
Broadfoot,  S.  K.     Motors,  Secondary  Batteries.     (Installation  Manuals 

Series) i2mo,  *o  75 

Broughton,  H.  H.    Electric  Cranes  and  Hoists *g  oo 

Brown,  G.     Healthy  Foundations.     (Science  Series  No.  80.) i6mo,  o  50 

Brown,  H.     Irrigation 8vo,  *5  oo 

Brown,  Wm.  N.     The  Art  of  Enamelling  on  Metal 12 mo,  *i  oo 

Handbook  on  Japanning  and  Enamelling i2mo,  *i  50 

House  Decorating  and  Painting i2mo,  *i  50 

History  of  Decorative  Art i2mo,  *i  25 

Dipping,  Burnishing,  Lacquering  and  Bronzing  Brass  Ware. . .  i2mo,  *i  oo 


6        D.   VAN   NOSTRAND   COMPANY'S   SHORT  TITLE   CATALOG 

Brown,  Wm.  N.    Workshop  Wrinkles 8vo,     *i  oo 

Browne,  R.  E.     Water  Meters.     (Science  Series  No.  81.) i6mo,  o  50 

Bruce,  E.  M.     Pure  Food  Tests i2mo,  *i  25 

Bruhns,  Dr.     New  Manual  of  Logarithms 8vo,  half  morocco,  2  oo 

Brunner,  R.     Manufacture  of  Lubricants,  Shoe  Polishes  and  Leather 

Dressings.     Trans,  by  C.  Salter 8vo,  *3  oo 

Buel,  R.  H.     Safety  Valves.     (Science  Series  No.  21.) i6mo,  o  50 

Bulman,  H.  F.,  and  Redmayne,  R.  S.  A.     Colliery  Working  and  Manage- 
ment  8vo,  6  oo 

Burgh,  N.  P.     Modern  Marine  Engineering 4to,  half  morocco,  10  oo 

BurstaU,  F.W.    Energy  Diagram  for  Gas.    With  Text 8vo,  150 

Diagram.     Sold  separately *i  oo 

Burt,  W.  A.     Key  to  the  Solar  Compass i6mo,  leather,  2  50 

Burton,  F.  G.     Engineering  Estimates  and  Cost  Accounts i2mo,  *i  50 

Buskett,  E.  W.     Fire  Assaying I2mo,  *i  25 

Byers,  H.  G.,  and  Knight,  H.  G.    Notes  on  Qualitative  Analysis 8vo,  *i  50 

Cain,  W.     Brief  Course  in  the  Calculus i2mo,  *i  75 

Elastic  Arches.     (Science  Series  No.  48.) i6mo,  o  50 

Maximum  Stresses.     (Science  Series  No.  38.) i6mo,  o  50 

Practical  Designing  Retaining  of  Walls.     (Science  Series  No.  3.) 

i6mo,  o  50 
Theory     of     Steel-concrete    Arches    and    of    Vaulted    Structures. 

(Science  Series  No.  42.) i6mo,  o  50 

Theory  of  Voussoir  Arches.     (Science  Series  No.  12.) i6mo,  o  50 

Symbolic  Algebra.     (Science  Series  No.  73.) i6mo,  o  50 

Campin,  F.     The  Construction  of  Iron  Roofs 8vo,  2  oo 

Carpenter,  F.  D.     Geographical  Surveying.     (Science  Series  No.  37.) .  i6mo, 

Carpenter,  R.  C.,  and  Diederichs,  H.     Internal  Combustion  Engines. 8vo,  *5  oo 

Carter,  E.  T.     Motive  Power  and  Gearing  for  Electrical  Machinery  .  .  8vo,  *5  oo 

Carter,  H.  A.     Ramie  (Rhea),  China  Grass i2mo,  *2  oo 

Carter,  H.  R.     Modern  Flax,  Hemp,  and  Jute  Spinning 8vo,  *3  oo 

Cathcart,  W.  L.     Machine  Design.     Part  I.  Fastenings 8vo,  *3  oo 

Cathcart,  W.  L.,  and  Chaff ee,  J.  I.     Elements  of  Graphic  Statics.  .  .  .  .8vo,  *3  oo 

Short  Course  in  Graphics i2mo,  i  50 

Caven,  R.  M.,  and  Lander,  G.  D.     Systematic  Inorganic  Chemistry.  i2mo,  *2  oo 

Chalkley,  A.  P.     Diesel  Engines 8vo,  *3  oo 

Chambers'  Mathematical  Tables 8vo,  i  75 

Charnock,  G.  F.     Workshop  Practice.     (Westminster  Series.) 8vo  (In  Press.) 

Charpentier,  P.     Timber 8vo,  *6  oo 

Chatley,  H.     Principles  and  Designs  of  Aeroplanes.    (Science   Series.) 

No.  126.) i6mo,  o  50 

How  to  Use  Water  Power i2mo,  *i  oo 

Child,   C.  D.    Electric  Arc 8vo,   *(In  Press.) 

Child,  C.  T.     The  How  and  Why  of  Electricity i2mo,  i  oo 

Christie,  W.  W.     Boiler- waters,  Scale,  Corrosion,  Foaming 8vo,  *3  oo 

Chimney  Design  and  Theory 8vo,  *3  oo 

Furnace  Draft.     (Science  Series  No.  123.) i6mo,  o  50 

Water:  Its  Purification  for  Use  in  the  Industries 8vo,  (In  Press.) 

Church's  Laboratory  Guide.     Rewritten  by  Edward  Kinch 8vo,  *2  50 

Clapperton,  G.     Practical  Papermaking 8vo,  2  50 


D.  VAN    NOSTRAND   COMPANY'S   SHORT   TITLE   CATALOG        7 

Clark,  A.  G.     Motor  Car  Engineering. 

Vol.  I.     Construction *3  oo 

Vol.  II.     Design (In  Press.) 

Clark,  C.  H.     Marine  Gas  Engines i2mo,  *i  50 

Clark,  D.  K.     Rules,  Tables  and  Data  for  Mechanical  Engineers 8vo,  5  oo 

Fuel:  Its  Combustion  and  Economy i2mo,  i  50 

The  Mechanical  Engineer's  Pocketbook i6mo,  2  oo 

—  Tramways:  Their  Construction  and  Working 8vo,  5  oo 

Clark,  J.  M.     New  System  of  Laying  Out  Railway  Turnouts i2mo,  i  oo 

Clausen-Thue,  W.     ABC  Telegraphic  Code.     Fourth  Edition i2mo,  *5  oo 

Fifth  Edition 8vo,  *7  oo 

The  A  i  Telegraphic  Code 8vo,  *y  50 

Cleemann,  T.  M.     The  Railroad  Engineer's  Practice i2mo,  *i  50 

Clerk,  D.,  and  Idell,  F.  E.     Theory  of  the  Gas  Engine.     (Science  Series 

No.  62.) i6mo,  o  50 

Clevenger,  S.  R.     Treatise    on   the    Method    of   Government    Surveying. 

i6mo,  morocco 2  50 

Clouth,  F.     Rubber,  Gutta-Percha,  and  Balata 8vo,  *s  oo 

Cochran,  J.    Treatise  on  Cement  Specifications 8vo,  (In  Press.). . 

Coffin,  J.  H.  C.     Navigation  and  Nautical  Astronomy i2mo,  *3  50 

Colburn,  Z.,  and  Thurston,  R.  H.     Steam  Boiler  Explosions.     (Science 

Series  No.  2.) i6mo,  o  50 

Cole,  R.  S.    Treatise  on  Photographic  Optics i2mo,  i  50 

Coles- Finch,  W.     Water,  Its  Origin  and  Use 8vo,  *5  oo 

Collins,  J.  E.     Useful  Alloys  and  Memoranda  for  Goldsmiths,  Jewelers. 

i6mo o  50 

Constantine,  E.     Marine  Engineers,  Their  Qualifications  and  Duties.    8vo,  *2  oo 

Coombs,  H.  A.     Gear  Teeth.     (Science  Series  No.  120.) i6mo,  o  50 

Cooper,  W.  R.     Primary  Batteries 8vo,  *4  oo 

"  The  Electrician  "  Primers . . .  ' 8vo,  *5  oo 

Part  I *i  50 

Part  II *2  50 

Part  HI *2  oo 

Copperthwaite,  W.  C.     Tunnel  Shields 4to,  *o  oo 

Corey,  H.  T.     Water  Supply  Engineering 8vo  (In  Press.) 

Corfield,  W.  H.     Dwelling  Houses.     (Science  Series  No.  50.) i6mo,  o  50 

Water  and  Water-Supply.     (Science  Series  No.  17.) i6mo,  o  50 

Cornwall,  H.  B.     Manual  of  Blow-pipe  Analysis 8vo,  *2  50 

Courtney,  C.  F.     Masonry  Dams 8vo,  3  50 

Cowell,  W.  B.     Pure  Air,  Ozone,  and  Water i2mo,  *2  oo 

Craig,  T.     Motion  of  a  Solid  in  a  Fuel.     (Science  Series  No.  49.) ....  i6mo,  o  50 

Wave  and  Vortex  Motion.     (Science  Series  No.  43.) i6mo,  o  50 

Cramp,  W.     Continuous  Current  Machine  Design 8vo,  *2  50 

Crocker,  F.  B.     Electric  Lighting.     Two  Volumes.     8vo. 

Vol.   I.     The  Generating  Plant 3  oo 

Vol.  II.     Distributing  Systems  and  Lamps 3  oo 

Crocker,  F.  B.,  and  Arendt,  M.     Electric  Motors 8vo,  *2  50 

Crocker,  F.  B.,  and  Wheeler,  S.  S.     The  Management  of  Electrical  Ma- 
chinery  i2mo,  *i  oo 

Cross,  C.  F.,  Bevan,  E.  J.,  and  Sindall,  R.  W.     Wood  Pulp  and  Its  Applica- 
tions.    (Westminster  Series.) 8vo,  *2  oo 


8        D.   VAN   NOSTRAND   COMPANY'S   SHORT  TITLE  CATALOG 

Crosskey,  L.  R.     Elementary  Perspective 8vo,  i  oo 

Crosskey,  L.  R.,  and  Thaw,  J.    Advanced  Perspective 8vo,  i  50 

Culley,  J.  L.      Theory  of  Arches.     (Science  Series  No.  87.) . . » i6mo,  o  50 

Davenport,  C.     The  Book.     (Westminster  Series.) 8vo,  *2  oo 

Da  vies,  D.  C.     Metalliferous  Minerals  and  Mining 8vo,  5  oo 

—  Earthy  Minerals  and  Mining 8vo,  5  oo 

Da  vies,  E.  H.     Machinery  for  Metalliferous  Mines 8vo,  8  oo 

Davies,  F.  H.     Electric  Power  and  Traction 8vo,  *2  oo 

Dawson,  P.     Electric  Traction  on  Railways 8vo,  *Q  oo 

Day,  C.     The  Indicator  and  Its  Diagrams I2mo,  *2  oo 

Deerr,  N.     Sugar  and  the  Sugar  Cane 8vo,  *8  oo 

Deite,  C.     Manual  of  Soapmaking.     Trans,  by  S.  T.  King 4to,  *5  oo 

De  la  Coux,  H.     The  Industrial  Uses  of  Water.     Trans,  by  A.  Morris. 

8vo,  *4  50 

Del  Mar,  W.  A.     Electric  Power  Conductors 8vo,  *2  oo 

Denny,  G.  A.     Deep-level  Mines  of  the  Rand 4to,  *io  oo 

Diamond  Drilling  for  Gold *5  oo 

De  Roos,  J.  D.  C.     Linkages.     (Science  Series  No.  47.) i6mo,  o  50 

Derr,  W.  L.     Block  Signal  Operation Oblong  i2mo,  *i  50 

Maintenance-of-Way  Engineering (In  Preparation.) 

Desaint,  A.     Three  Hundred  Shades  and  How  to  Mix  Them 8vo,  *io  oo 

De  Varona,  A.     Sewer  Gases.     (Science  Series  No.  55.).. i6mo,  o  50 

Devey,  R.  G.     Mill  and  Factory  Wiring.     (Installation  Manuals  Series.) 

lamo,  *i  oo 

Dibdin,  W.  J.     Public  Lighting  by  Gas  and  Electricity 8vo,  *8  oo 

Purification  of  Sewage  and  Water 8vo,  6  50 

Dichmann,  Carl.    Basic  Open-Hearth  Steel  Process i2mo,  *3  50 

Dieterich,  K.     Analysis  of  Resins,  Balsams,  and  Gum  Resins 8vo,  *3  oo 

Dinger,  Lieut.  H.  C.     Care  and  Operation  of  Naval  Machinery 12 mo,  *2  oo 

Dixon,  D.  B.     Machinist's  and  Steam  Engineer's  Practical  Calculator. 

i6mo,  morocco,  i  25 

Doble,  W.  A.     Power  Plant  Construction  on  the  Pacific  Coast  (In  Press.) 
Dodd,  G.     Dictionary    of   Manufactures,    Mining,    Machinery,    and    the 

Industrial  Arts i2mo,  i  50 

Dorr,  B.  F.     The  Surveyor's  Guide  and  Pocket  Table-book. 

i6mo,  morocco,  2  oo 

Down,  P.  B.     Handy  Copper  Wire  Table i6mo,  *i  oo 

Draper,  C.  H.     Elementary  Text-book  of  Light,  Heat  and  Sound. . .  i2mo,  i  oo 

Heat  and  the  Principles  of  Thermo-dynamics i2mo,  i  50 

Duckwall,  E.  W.     Canning  and  Preserving  of  Food  Products 8vo,  *5  oo 

Dumesny,  P.,  and  Noyer,  J.     Wood  Products,  Distillates,  and  Extracts. 

8vo,  *4  50 
Duncan,  W.  G.,  and  Penman,  D.     The  Electrical  Equipment  of  Collieries. 

8vo,  *4  oo 
Dunstan,rA.  E.,  and  Thole,  F.  B.  T.    Textbook  of  Practical  Chemistry. 

i2mo,  *i  40 

Duthie,  A.  L.     Decorative  Glass  Processes.     (Westminster  Series.).  .8vo,  *2  oo 

Dwight,  H.  B.    Transmission  Line  Formulas .8vo,   (In  Press.) 

Dyson,  S.  S.     Practical  Testing  of  Raw  Materials 8vo,  *5  oo 

Dyson,  S.  S.,  and  Clarkson,  S.  S.     Chemical  Works 8vo,  *7  50 


D.  VAN   NOSTRAND   COMPANY'S   SHORT  TITLE  CATALOG        9 

Eccles,  R.  G.,  and  Duckwall,  E.  W.     Food  Preservatives 8vo,  paper  o  50 

Eddy,  H.  T.     Researches  in  Graphical  Statics 8vo,  i  50 

Maximum  Stresses  under  Concentrated  Loads 8vo,  i  50 

Edgcumbe,  K.     Industrial  Electrical  Measuring  Instruments 8vo,  *2  50 

Eissler,  M.     The  Metallurgy  of  Gold 8vo>  7  50 

The  Hydrometallurgy  of  Copper 8vo,  *4  50 

The  Metallurgy  of  Silver 8vo,  4  oo 

The  Metallurgy  of  Argentiferous  Lead 8vo,  5  oo 

Cyanide  Process  for  the  Extraction  of  Gold 8vo,  3  oo 

A  Handbook  on  Modern  Explosives 8vo,  5  oo 

Ekin,  T.  C.     Water  Pipe  and  Sewage  Discharge  Diagrams folio,  *3  oo 

Eliot,  C.  W.,  and  Storer,  F.  H.     Compendious  Manual  of  Qualitative 

Chemical  Analysis I2mo,  *i  25 

Elliot,  Major  G.  H.     European  Light-house  Systems 8vo,  5  oo 

Ennis,  Win.  D.     Linseed  Oil  and  Other  Seed  Oils 8vo,  *4  oo 

Applied  Thermodynamics 8vo  *4  50 

Flying  Machines  To-day i2mo,  *i  50 

Vapors  for  Heat  Engines i2mo,  *i  oo 

Erfurt,  J.     Dyeing  of  Paper  Pulp.     Trans,  by  J.  Hubner 8vo,  *7  50 

Erskine-Murray,  J.     A  Handbook  of  Wireless  Telegraphy 8vo,  *3  50 

Evans,  C.  A.     Macadamized  Roads (In  Press.) 

Ewing,  A.  J.     Magnetic  Induction  in  Iron 8vo,  *4  oo 

Fairie,  J.     Notes  on  Lead  Ores i2mo,  *i  oo 

Notes  on  Pottery  Clays I2mo,  *i  50 

Fairley,  W.,  and  Andre,  Geo.  J.     Ventilation  of  Coal  Mines.     (Science 

Series  No.  58.) i6mo,  o  50 

Fairweather,  W.  C.     Foreign  and  Colonial  Patent  Laws 8vo,  *3  oo 

Fanning,  J.  T.     Hydraulic  and  Water-supply  Engineering 8vo,  *5  oo 

Fauth,  P.      The  Moon  in  Modern  Astronomy.     Trans,  by  J.  McCabe. 

8vo,  *2  oo 

Fay,  I.  W.     The  Coal-tar  Colors 8vo,  *4  oo 

Fernbach,  R.  L.     Glue  and  Gelatine 8vo,  *3  oo 

Chemical  Aspects  of  Silk  Manufacture i2mo,  *i  oo 

Fischer,  E.     The  Preparation  of  Organic  Compounds.     Trans,  by  R.  V. 

Stanford 12010,  *i  25 

Fish,  J.  C.  L.     Lettering  of  Working  Drawings Oblong  8vo,  i  oo 

Fisher,  H.  K.  C.,  and  Darby,  W.  C.     Submarine  Cable  Testing 8vo,  *3  50 

Fiske,  Lieut.  B.  A.     Electricity  in  Theory  and  Practice 8vo,  2  50 

Fleischmann,  W.    The  Book  of  the  Dairy.  Trans,  by  C.  M.  Aikman.   8vo,  4  oo 
Fleming,  J.  A.     The  Alternate-current  Transformer.     Two  Volumes.    8vo. 

Vol.    I.     The  Induction  of  Electric  Currents *5  oo 

Vol.  II.     The  Utilization  of  Induced  Currents *5  oo 

Propagation  of  Electric  Currents 8vo,  *3  oo 

Centenary  of  the  Electrical  Current 8vo,  *o  50 

Electric  Lamps  and  Electric  Lighting 8vo,  *3  oo 

Electrical  Laboratory  Notes  and  Forms 4to,  *5  oo 

A  Handbook  for  the  Electrical  Laboratory  and  Testing  Room.     Two 

Volumes 8vo,  each,  *5  oo 

Fluery,  H.     The  Calculus  Without  Limits  or  Infinitesimals.     Trans,  by 
C.  0.  Mailloux (In  Press.) 


JLU     D.   VAN  NOSTRAND  COMPANY'S  SHORT  TITLE   CATALOG 

jpiynn,  P.  J.     Flow  of  Water.     (Science  Series  No.  84.) i6mo,  o  50 

Hydraulic  Tables.     (Science  Series  No.  66.) i6mo,  o  50 

Foley,  N.     British  and  American  Customary  and  Metric  Measures .  .  folio,  *3  oo 
Foster,  H.  A.     Electrical  Engineers'  Pocket-book.     (Sixth  Edition.) 

i2mo,  leather,  5  oo 

Engineering  Valuation  of  Public  Utilities  and  Factories 8vo,  *3  oo 

Foster,  Gen.  J.  G.     Submarine  Blasting  in  Boston  (Mass.)  Harbor. .  .  .  4to,  3  50 

Fowle,  F.  F.     Overhead  Transmission  Line  Crossings i2mo,  *i  50 

The  Solution  of  Alternating  Current  Problems 8vo  (In  Press.) 

Fox,  W.  G.     Transition  Curves.     (Science  Series  No.  1 10.) i6mo,  o  50 

Fox,  W.,  and  Thomas,  C.  W.     Practical  Course  in  Mechanical  Draw- 
ing  iimo,  i  25 

Foye,  J.  C.     Chemical  Problems.     (Science  Series  No.  69.) i6mo,  o  50 

Handbook  of  Mineralogy.     (Science  Series  No.  86.) i6mo,  o  50 

Francis,  J.  B.     Lowell  Hydraulic  Experiments 4to,  15  oo 

Freudemacher,    P.    W.     Electrical    Mining    Installations.     (Installation 

Manuals  Series  ) i2mo,  *i  oo 

Frith,  J.    Alternating  Current  Design 8vo,  *2  oo 

Fritsch,  J.     Manufacture  of  Chemical  Manures.    Trans,  by  D.  Grant. 

8vo,  *4  oo 

Frye,  A.  I.     Civil  Engineers'  Pocket-book i2mo,  leather, 

Fuller,  G.  W.      Investigations  into  the  Purification  of  the   Ohio  River. 

4to.  *io  oo 

Furnell,  J.     Paints,  Colors,  Oils,  and  Varnishes 8vo,  *i  oo 

Gairdner,  J.  W.  I.    Earthwork 8vo,   (In  Press.) 

Gant,  L.  W.     Elements  of  Electric  Traction 8vo,  *2  50 

Garforth,  W.  E.     Rules  for  Recovering  Coal  Mines  after  Explosions  and 

Fires i2mo,  leather,  i  50 

Gaudard,  J.     Foundations.     (Science  Series  No.  34.) i6mo,  o  50 

Gear,  H.  B.,  and  Williams,  P.  F.     Electric  Central  Station  Distribution 

Systems 8vo,  *3  oo 

Geerligs,  H.  C.  P.     Cane  Sugar  and  Its  Manufacture 8vo,  *5  oo 

Geikie,  J.     Structural  and  Field  Geology 8vo,  *4  oo 

Gerber,  N.   Analysis  of  Milk,  Condensed  Milk,  and  Infants' Milk-Food.    8vo,  i  25 
Gerhard,  W.  P.     Sanitation,  Watersupply  and  Sewage  Disposal  of  Country 

Houses i2mo,  *2  oo 

Gas  Lighting.     (Science  Series  No.  in.) i6mo,  o  50 

Household  Wastes.     (Science  Series  No.  97.) i6mo,  o  50 

House  Drainage.     (Science  Series  No.  63.) i6mo,  o  50 

Sanitary  Drainage  of  Buildings.     (Science  Series  No.  93.) i6mo,  o  50 

Gerhardi,  C.  W.  H.     Electricity  Meters 8vo,  *4  oo 

Geschwind,   L.     Manufacture   of   Alum  and  Sulphates.     Trans,   by   C. 

Salter 8vo,  *5  oo 

Gibbs,  W.  E.     Lighting  by  Acetylene i2mo,  *i  50 

Physics  of  Solids  and  Fluids.     (Carnegie  Technical  School's  Text- 
books.)   *i  50 

Gibson,  A.  H.     Hydraulics  and  Its  Application 8vo,  *5  oo 

Water  Hammer  in  Hydraulic  Pipe  Lines i2mo,  *2  oo 

Gilbreth,  F.  B.     Motion  Study i2mo,  *2  oo 

Primer  of  Scientific  Management i2mo,  *i  oo 


D.  VAN   NOSTRAND   COMPANY'S   SHORT  TITLE  CATALOG      11 

Gillmore,  Gen.  Q.  A.     Limes,  Hydraulic  Cements  ard  Mortars 8vo,  4  oo 

Roads,  Streets,  and  Pavements - .  i2mo,  2  oo 

Golding,  H.  A.     The  Theta-Phi  Diagram i2mo,  *i  25 

Goldschmidt,  R.     Alternating  Current  Commutator  Motor 8vo,  *3  oo 

Goodchild,  W.     Precious  Stones.     (Westminster  Series.) 8vo,  *2  oo 

Goodeve,  T.  M.     Textbook  on  the  Steam-engine I2mo,  2  oo 

Gore,  G.     Electrolytic  Separation  of  Metals 8vo,  *3  50 

Gould,  E.  S.     Arithmetic  of  the  Steam-engine i2mo,  i  oo 

Calculus.     (Science  Series  No.  112.) i6mo,  o  50 

High  Masonry  Dams.     (Science  Series  No.  22.) i6mo,  o  50 

Practical  Hydrostatics  and  Hydrostatic  Formulas.     (Science  Series 

No.  117.) i6mo,  o  50 

Grant,  J.     Brewing  and  Distilling.     (Westminster  Series.)  8vo  (In  Press.) 

Gratacap,  L.  P.    A  Popular  Guide  to  Minerals 8vo  (In  Press.) 

Gray,  J.     Electrical  Influence  Machines i2mo,  2  oo 

Marine   Boiler   Design i2mo,    (In  Press.) 

Greenhill,  G.     Dynamics  of  Mechanical  Flight 8vo,  (In  Press.) 

Greenwood,  E.     Classified  Guide  to  Technical  and  Commercial  Books.  8vo,  *3  oo 

Gregorius,  R.     Mineral  Waxes.     Trans,  by  C.  Salter i2mo,  *3  oo 

Griffiths,  A.  B.     A  Treatise  on  Manures i2mo,  3  oo 

Dental  Metallurgy 8vo,  *3  50 

Gross,  E.     Hops 8vo,  *4  50 

Grossman,  J.     Ammonia  and  Its  Compounds I2mo,  *i  25 

Groth,  L.  A.     Welding  and  Cutting  Metals  by  Gases  or  Electricity ....  8vo,  *3  oo 

Grover,  F.     Modern  Gas  and  Oil  Engines 8vo,  *2  oo 

Gruner,  A.     Power-loom  Weaving 8vo,  *3  oo 

Giildner,  Hugo.     Internal  Combustion  Engines.     Trans,  by  H.  Diederichs, 

4to,  *io  06 

Gunther,  C.  0.     Integration I2mo,  *i  25 

Gurden,  R.  L.     Traverse  Tables folio,  half  morocco,  *7  50 

Guy,  A.  E.     Experiments  on  the  Flexure  of  Beams 8vo,  *i  25 

Haeder,    H.      Handbook   on    the    Steam-engine.      Trans,  by  H.  H.  P. 

Powles. i2mo,  3  oo 

Hainbach,  R.     Pottery  Decoration.     Trans,  by  C.  Slater i2mo,  *3  oo 

Haenig,  A.     Emery  and  Emery  Industry 8vo,  (In  Press.) 

Hale,  W.  J.     Calculations  of  General  Chemistry i2mo,  *i  oo 

Hall,  C.  H.     Chemistry  of  Paints  and  Paint  Vehicles I2mo,  *2  oo 

Hall,  R.  H..    Governors  and  Governing  Mechanism i2mo,  *2  oo 

Hall,  W.  S.     Elements  of  the  Differential  and  Integral  Calculus 8vo,  *2  25 

—  Descriptive  Geometry 8vo  volume  and  a  4to  atlas,  *3  50 

Haller,  G.  F.,  and  Cunningham,  E.  T.     The  Tesla  Coil I2mo,  *i  25 

Halsey,  F.  A.     Slide  Valve  Gears i2mo,  i  50 

The  Use  of  the  Slide  Rule.     (Science  Series  No.  114.) i6mo,  o  50 

-  Worm  and  Spiral  Gearing.     (Science  Series  No.  116.) i6mo,  o  50 

Hamilton,  W.  G.     Useful  Information  for  Railway  Men i6mo,  i  oo 

Hammer,  W.  J.     Radium  and  Other  Radio-active  Substances 8vo,  *i  oo 

Hancock,  H.     Textbook  of  Mechanics  and  Hydrostatics 8vo,  i  50 

Hardy,  E.     Elementary  Principles  of  Graphic  Statics I2mo,  *i  50 

Harrison,  W.  B.     The  Mechanics'  Tool-book i2mo,  i  50 

Hart,  J.  W.     External  Plumbing  Work 8vo,  *3  oo 


12     D.  VAN   NOSTRAND  COMPANY'S  SHORT  TITLE   CATALOG 

Hart,  J.  W.    Hints  to  Plumbers  on  Joint  Wiping 8vo,  *3  oo 

Principles  of  Hot  Water  Supply 8vo,  *3  oo 

Sanitary  Plumbing  and  Drainage 8vo,  *3  oo 

Haskins,  C.  H.     The  Galvanometer  and  Its  Uses i6mo,  i  50 

Hatt,  J.  A.  H.     The  Colorist square  i2mo,  *i  50 

Hausbrand,  E.     Drying  by  Means  of  Air  and  Steam.     Trans,  by  A.  C. 

Wright i2mo,  *2  oo 

Evaporating,  Condensing  and  Cooling  Apparatus.     Trans,  by  A.  C. 

Wright 8vo,  *s  oo 

Hausner,  A.     Manufacture  of  Preserved  Foods  and  Sweetmeats.     Trans. 

by  A.  Morris  and  H.  Robson 8vo,  *3  oo 

Hawke,  W.  H.     Premier  Cipher  Telegraphic  Code 4to,  *5  oo 

100,000  Words  Supplement  to  the  Premier  Code 4to,  *5  oo 

Hawkesworth,  J.     Graphical  Handbook  for  Reinforced  Concrete  Design. 

4to,  *2  50 

Hay,  A.     Alternating  Currents 8vo,  *2  50 

Electrical  Distributing  Networks  and  Distributing  Lines 8vo,  *3  50 

Continuous  Current  Engineering 8vo,  *2  50 

Heap,  Major  D.  P.     Electrical  Appliances 8vo,  2  oo 

Heaviside,  0.     Electromagnetic  Theory.     Two  Volumes 8vo,  each,  *5  oo 

Heck,  R.  C.  H.    The  Steam  Engine  and  Turbine 8vo,  *5  oo 

Steam-Engine  and  Other  Steam  Motors.    Two  Volumes. 

Vol.    I.     Thermodynamics  and  the  Mechanics 8vo,  *3  50 

Vol.  II.     Form,  Construction,  and  Working 8vo,  *s  oo 

•  Notes  on  Elementary  Kinematics 8vo,  boards,  *i  oo 

Graphics  of  Machine  Forces 8vo,  boards,  *i  oo 

Hedges,  K.     Modern  Lightning  Conductors 8vo,  3  oo 

Heermann,  P.     Dyers'  Materials.     Trans,  by  A.  C.  Wright i2mo,  *2  50 

Hellot,  Macquer  and  D'Apligny.     Art  of  Dyeing  Wool,  Silk  and  Cotton. 

8vo,  *2  oo 

Henrici,  0.     Skeleton  Structures 8vo,  i  50 

Hering,  D.  W.    Essentials  of  Physics  for  College  Students 8vo,  *i  60 

Hering-Shaw,  A.     Domestic  Sanitation  and  Plumbing.     Two  Vols. .  .8vo,  *5  oo 

Elementary  Science 8vo,  *2  oo 

Herrmann,  G.     The  Graphical  Statics  of  Mechanism.     Trans,  by  A.  P. 

Smith 12010,  2  oo 

Herzfeld,  J.     Testing  of  Yarns  and  Textile  Fabrics 8vo,  *3  50 

Hildebrandt,  A.     Airships,  Past  and  Present 8vo,  *3  50 

Hildenbrand,  B.  W.     Cable-Making.     (Science  Series  No.  32.) i6mo,  o  50 

Hilditch,  T.  P.     A  Concise  History  of  Chemistry i2mo,  *i  25 

Hill,  J.  W.     The  Purification  of  Public  Water  Supplies.     New  Edition. 

(In  Press.) 

Interpretation  of  Water  Analysis (In  Press.) 

Hiroi,  I.     Plate  Girder  Construction.     (Science  Series  No.  95.) i6mo,  o  50 

Statically-Indeterminate  Stresses i2mo,  *2  oo 

Hirshfeld,  C.  F.     Engineering  Thermodynamics.     (Science  Series  No.  45.) 

i6mo,  o  50 

Hobart,  H.  M.     Heavy  Electrical  Engineering 8vo,  *4  50 

Design  of  Static  Transformers izmo,  *2  oo 

Electricity 8vo,  *2  oo 

Electric  Trains  .  .                                                8vo,  *2  50 


D.  VAN  NOSTRAND  COMPANY'S  SHORT  TITLE  CATALOG      13 

Hobart,  H.  M.    Electric  Propulsion  of  Ships 8vo,  *2  oo 

Hobart,   J.   F.    Hard   Soldering,   Soft   Soldering   and    Brazing. i2mo, 

(In  Press.) 

Hobbs,  W.  R.  P.     The  Arithmetic  of  Electrical  Measurements i2mo,  o  50 

Hoff,  J.  N.     Paint  and  Varnish  Facts  and  Formulas i2ino,  *i  50 

Hoff,  Com.  W.  B.     The  Avoidance  of  Collisions  at  Sea.  .  .  i6mo,  morocco,  o  75 

Hole,  W.     The  Distribution  of  Gas 8vo,  *7  50 

Holley,  A.  L.     Railway  Practice folio,  12  oo 

Holmes,  A.  B.     The  Electric  Light  Popularly  Explained  ....  i2mo,  paper,  o  50 

Hopkins,  N.  M.     Experimental  Electrochemistry 8vo,  *3  oo 

—  Model  Engines  and  Small  Boats 12010,  i  25 

Hopkinson,  J.     Shoolbred,  J.  N.,  and  Day,  R.  E.     Dynamic  Electricity. 

(Science  Series  No.  71.) i6mo,  o  50 

Homer,  J.     Engineers'  Turning 8vo,  *3  50 

Metal  Turning i2mo,  i  50 

-  Toothed  Gearing i2mo,  2  25 

Houghton,  C.  E.     The  Elements  of  Mechanics  of  Materials i2mo,  *2  oo 

Houllevigue,  L.    The  Evolution  of  the  Sciences 8vo,  *2  oo 

Howe,  G.     Mathematics  for  the  Practical  Man i2mo,  *i  25 

Howorth,  J.     Repairing  and  Riveting  Glass,  China  and  Earthenware. 

8vo,  paper,  *o  50 

Hubbard,  E.     The  Utilization  of  Wood- waste 8vo,  *2  50 

Hiibner,  J.    Bleaching  and  Dyeing  of  Vegetable  and  Fibrous  Materials 

(Outlines  of  Industrial  Chemistry) 8vo,  (In  Press.) 

Hudson,  O.  F.    Iron  and  Steel.     (Outlines  of  Industrial  Chemistry.) 

8vo,  (In  Press.) 

Humper,  W.     Calculation  of  Strains  in  Girders i2mo,  2  50 

Humphreys,  A.  C.     The  Business  Features  of  Engineering  Practice .  8vo,  *i  25 

Hunter,  A.    Bridge  Work 8vo,  (In  Press.) 

Hurst,  G.  H.     Handbook  of  the  Theory  of  Color 8vo,  *2  50 

—  Dictionary  of  Chemicals  and  Raw  Products 8vo,  *3  oo 

Lubricating  Oils,  Fats  and  Greases 8vo,  *4  oo 

—  Soaps 8vo,  *s  oo 

-  Textile  Soaps  and  Oils 8vo,  *2  50 

Hurst,  H.  E.,  and  Lattey,  R.  T.     Text-book  of  Physics 8vo,  *3  oo 

Hutchinson,  R.  W.,  Jr.     Long  Distance  Electric  Power  Transmission. 

i2mo,  *3  oo 

Hutchinson,  R.  W.,  Jr.,  and  Ihlseng,  M.  C.     Electricity  in  Mining.  .  i2mo, 

(In  Press) 

Hutchinson,  W.  B.  •  Patents  and  How  to  Make  Money  Out  of  Them.  i2mo,  i  25 

Hutton,  W.  S.     Steam-boiler  Construction 8vo,  6  oo 

Practical  Engineer's  Handbook 8vo,  7  oo 

—  The  Works'  Manager's  Handbook 8vo,  6  oo 

Hyde,  E.  W.     Skew  Arches.     (Science  Series  No.  15.) i6mo,  o  50 

Induction  Coils.     (Science  Series  No.  53.) i6mo,  o  50 

Ingle,  H.     Manual  of  Agricultural  Chemistry 8vo,  *3  oo 

Innes,  C.  H.     Problems  in  Machine  Design i2mo,  *2  oo 

Air  Compressors  and  Blowing  Engines I2mo,  *2  oo 

Centrifugal  Pumps i2mo,  *2  oo 

The  Fan I2mo,  *2  oo 


14      D.  VAN   NOSTRAND   COMPANY'S  SHORT  TITLE   CATALOG 

Isherwood,  B.  F.     Engineering  Precedents  for  Steam  Machinery 8vo,  2  50 

Ivatts,  E.  B.     Railway  Management  at  Stations 8vo,  *2  50 

Jacob,  A.,  and  Gould,  E.  S.     On  the  Designing  and  Construction  of 

Storage  Reservoirs.     (Science  Series  No.  6.) i6mo,  o  50 

Jamieson,  A.     Text  Book  on  Steam  and  Steam  Engines 8vo,  3  oo 

Elementary  Manual  on  Steam  and  the  Steam  Engine i2mo,  i  50 

Jannettaz,  E.     Guide  to  the  Determination  of  Rocks.     Trans,  by  G.  W. 

Plympton I2mo,  i  50 

Jehl,  F.     Manufacture  of  Carbons 8vo,  *4  oo 

Jennings,  A.  S.     Commercial  Paints  and  Painting.     (Westminster  Series.) 

8vo  (In  Press.) 

Jennison,  F.  H.     The  Manufacture  of  Lake  Pigments 8vo,  *3  oo 

Jepson,  G.     Cams  and  the  Principles  of  their  Construction 8vo,  *i  50 

Mechanical  Drawing 8vo  (In  Preparation.) 

Jockin,  W.     Arithmetic  of  the  Gold  and  Silversmith i2mo,  *i  oo 

Johnson,  G.  L.     Photographic  Optics  and  Color  Photography 8vo,  *3  oo 

Johnson,  J.  H.      Arc  Lamps  and  Accessory  Apparatus.     (Installation 

Manuals  Series.) i2mo,  *o  75 

Johnson,    T.    M.      Ship    Wiring    and    Fitting.       (Installation    Manuals 

Series) I2mo,  *o  75 

Johnson,  W.  H.     The  Cultivation  and  Preparation  of  Para  Rubber. .  .8vo,  *3  oo 

Johnson,  W.  McA.     The  Metallurgy  of  Nickel (In  Preparation.) 

Johnston,  J.  F.  W.,  and  Cameron,  C.     Elements  of  Agricultural  Chemistry 

and  Geology I2mo,  2  60 

Joly,  J.     Raidoactivity  and  Geology I2mo,  *3  oo 

Jones,  H.  C.     Electrical  Nature  of  Matter  and  Radioactivity i2mo,  *2  oo 

Jones,  M.  W.     Testing  Raw  Materials  Used  in  Paint i2mo,  *2  oo 

Jones,  L.,  and  Scard,  F.  I.     Manufacture  of  Cane  Sugar 8vo,  *5  oo 

Jordan,  L.  C.    Practical  Railway  Spiral i2mo,  Leather,  (In  Press.) 

Joynson,  F.  H.     Designing  and  Construction  of  Machine  Gearing. .  .  .8vo,  2  oo 

Juptner,  H.  F.  V.     Siderology :  The  Science  of  Iron 8vo,  *s  oo 

Kansas  City  Bridge : 4to,  6  oo 

Kapp,  G.     Alternate  Current  Machinery.     (Science  Series  No.  96.) .  i6mo,  o  50 

Electric  Transmission  of  Energy i2mo,  3  50 

Keim,  A.  W.     Prevention  of  Dampness  in  Buildings 8vo,  *2  oo 

Keller,  S.  S.     Mathematics  for  Engineering  Students.     i2mo,  half  leather. 

Algebra  and  Trigonometry,  with  a  Chapter  on  Vectors *i  75 

Special  Algebra  Edition *i  oo 

Plane  and  Solid  Geometry *i  25 

Analytical  Geometry  and  Calculus *2  oo 

Kelsey,  W.  R.     Continuous-current  Dynamos  and  Motors 8vo,  *2  50 

Kemble,  W.  T.,  and  Underbill,  C.  R.     The  Periodic  Law  and  the  Hydrogen 

Spectrum 8vo,  paper,  *o  50 

Kemp,  J.  F.     Handbook  of  Rocks 8vo,  *i  50 

Kendall,  E.    Twelve  Figure  Cipher  Code 4*0,  *i2  50 

Kennedy,  A.  B.  W.,  and  Thurston,  R.  H.     Kinematics  of  Machinery. 

(Science  Series  No.  54.) i6mo,  o  50 

Kennedy,  A.  B.  W.,  Unwin,  W.  C.,  and  Idell,  F.  E.     Compressed  Air. 

(Science  Series  No.  106.) i6mo,  o  50 


D.   VAN   N( STRAND   COMPANY'S   SHORT  TITLE   CATALOG      15 

Kennedy,  R.     Modern  Engines  and  Power  Generators.     Six  Volumes.   4to,  15  oo 

Single  Volumes each,  3  oo 

Electrical  Installations.     Five  Volumes 4to,  15  oo 

Single  Volumes each,  3  50 

Flying  Machines;  Practice  and  Design i2mo,  *2  oo 

Principles  of  Aeroplane  Construction 8vo,  *i  50 

Kennelly,  A.  E.     Electro-dynamic  Machinery 8vo,  i  50 

Kent,  W.     Strength  of  Materials.     (Science  Series  No.  41.) i6mo,  o  50 

Kershaw,  J.  B.  C.     Fuel,  Water  and  Gas  Analysis 8vo,  *2  50 

Electrometallurgy.     (Westminster  Series.) 8vo,  *2  oo 

—  The  Electric  Furnace  in  Iron  and  Steel  Production i2mo,  *i  50 

Kinzbrunner,  C.     Alternate  Current  Windings 8vo,  *i  50 

—  Continuous  Current  Armatures 8vo,  *i  50 

—  Testing  of  Alternating  Current  Machines ...  8vo,  *2  oo 

Kirkaldy,  W.  G.     David  Kirkaldy's  System  of  Mechanical  Testing 4to,  10  oo 

Kirkbride,  J.     Engraving  for  Illustration 8vo,  *i  50 

Kirkwood,  J.  P.     Filtration  of  River  Waters 4to,  7  50 

Klein,  J.  F.     Design  of  a  High-speed  Steam-engine 8vo,  *5  oo 

Physical  Significance  of  Entropy 8vo,  *i  50 

Kleinhans,  F.  B.     Boiler  Construction 8vo,  3  oo 

Knight,  R.-Adm.  A.  M.     Modern  Seamanship 8vo,  *7  50 

Half  morocco *9  oo 

Knox,  W.  F.     Logarithm  Tables (In  Preparation.} 

Knott,  C.  G.,  and  Mackay,  J.  S.     Practical  Mathematics 8vo,  2  oo 

Koester,  F.     Steam-Electric  Power  Plants 4to,  *s  oo 

—  Hydroelectric  Developments  and  Engineering 4to,  *5  oo 

Koller,  T.     The  Utilization  of  Waste  Products 8vo,  *3  50 

—  Cosmetics 8vo,  *2  50 

Kretchmar,  K.     Yarn  and  Warp  Sizing 8vo,  *4  oo 

Krischke,  A.     Gas  and  Oil  Engines i2mo,  *i  25 

Lambert,  T.     Lead  and  its  Compounds 8vo,  *3  50 

—  Bone  Products  and  Manures 8vo,  *3  oo 

Lamborn,  L.  L.     Cottonseed  Products 8vo,  *3  oo 

Modern  Soaps,  Candles,  and  Glycerin 8vo,  *7  50 

Lamprecht,  R.     Recovery  Work  After  Pit  Fires.     Trans,  by  C.  Salter .  .  8vo,  *4  oo 
Lanchester,  F.  W.     Aerial  Flight.     Two  Volumes.     8vo. 

Vol.  I.  Aerodynamics *6  oo 

Aerial  Flight.  Vol.  II.  Aerodonetics *6  oo 

Larner,  E.  T.  Principles  of  Alternating  Currents i2mo,  *i  25 

Larrabee,  C.  S.  Cipher  and  Secret  Letter  and  Telegraphic  Code i6mo,  o  60 

La  Rue,  B.  F.  Swing  Bridges.  (Science  Series  No.  107.) i6mo,  o  50 

Lassar-Cohn,  Dr.  Modern  Scientific  Chemistry.  Trans,  by  M.  M.  Patti- 

son  Muir i2mo,  *2  oo 

Latimer,  L.  H.,  Field,  C.  J.,  and  Howell,  J.  W.  Incandescent  Electric 

Lighting.  (Science  Series  No.  57.) i6mo,  o  50 

Latta,  M.  N.  Handbook  of  American  Gas-Engineering  Practice 8vo,  *4  50 

American  Producer  Gas  Practice 4to,  *6  oo 

Leask,  A.  R.  Breakdowns  at  Sea i2mo,  2  oo 

Refrigerating  Machinery i2mo,  2  oo 

Lecky,  S.  T.  S.  "  Wrinkles  "  in  Practical  Navigation 8vo,  *8 


16      D.  VAN   NOSTRAND   COMPANY'S  SHORT  TITLE   CATALOG 

Le  Doux,  M.     Ice-Making  Machines.     (Science  Series  No.  46.) ....  i6mo,  o  50 

Leeds,  C.  C.     Mechanical  Drawing  foi  Trade  Schools oblong  4to, 

High  School  Edition *i  25 

Machinery  Trades  Edition *2  oo 

Lefe"vre,  L.     Architectural  Pottery.      Trans,  by  H.  K.  Bird  and  W.  M. 

Binns 4to,  *7  50 

Lehner,  S.     Ink  Manufacture.     Trans,  by  A.  Morris  and  H.  Robson  . .  8vo,  *2  50 

Lemstrom,  S.     Electricity  in  Agriculture  and  Horticulture 8vo,  *i  50 

Le  Van,  W.  B.     Steam-Engine  Indicator.     (Science  Series  No.  78.) .  i6mo,  o  50 

Lewes,  V.  B.     Liquid  and  Gaseous  Fuels.     (Westminster  Series.). . .  .8vo,  *2  oo 

Lewis,  L.  P.    Railway  Signal  Engineering 8vo,  *3  50 

Lieber,  B.  F.     Lieber's  Standard  Telegraphic  Code Svo,  *io  oo 

Code.     German  Edition Svo,  *io  oo 

—  Spanish  Edition Svo,  *io  oo 

French  Edition Svo,  *io  oo 

Terminal  Index Svo,  *2  50 

Lieber's  Appendix folio,  *is  oo 

—  Handy  Tables 4to,  *2  50 

Bankers  and  Stockbrokers'  Code  and  Merchants  and  Shippers'  Blank 

Tables Svo,  *i5  oo 

100,000,000  Combination  Code Svo,  *io  oo 

Engineering  Code Svo,  *I2  50 

Livermore,  V.  P.,  and  Williams,  J.     How  to  Become  a  Competent  Motor- 
man i2mo,  *i  oo 

Livingstone,  R.     Design  and  Construction  of  Commutators Svo,  *2  25 

Lobben,  P.     Machinists'  and  Draftsmen's  Handbook         Svo,  2  50 

Locke,  A.  G.  and  C.  G.     Manufacture  of  Sulphuric  Acid Svo,  10  oo 

Lockwood,  T.  D.     Electricity,  Magnetism,  and  Electro-telegraph  ....  Svo,  2  50 

Electrical  Measurement  and  the  Galvanometer i2mo,  o  75 

Lodge,  0.  J.     Elementary  Mechanics I2mo,  i  50 

—  Signalling  Across  Space  without  Wires Svo,  *2  oo 

Loewenstein,  L.  C.,  and  Crissey,  C.  P.     Centrifugal  Pumps *4  50 

Lord,  R.  T.     Decorative  and  Fancy  Fabrics Svo,  *3  50 

Loring,  A.  E.     A  Handbook  of  the  Electromagnetic  Telegraph i6mo,  o  50 

Handbook.     (Science  Series  No.  39.) i6mo,  o  50 

Low,  D.  A.    Applied  Mechanics  (Elementary) i6mo,  o  So 

Lubschez,  B    J.     Perspective (In  Press.) 

Lucke,  C.  E.*    Gas  Engine  Design Svo,  *3  oo 

Power  Plants:  Design,  Efficiency,  and  Power  Costs.  2  vols.  (In  Preparation.) 

Lunge,  G.     Coal-tar  and  Ammonia.     Two  Volumes Svo,  *is  oo 

Manufacture  of  Sulphuric  Acid  and  Alkali.     Four  Volumes Svo, 

Vol.     I.     Sulphuric  Acid.     In  two  parts *i5  oo 

Vol.   II.     Salt  Cake,  Hydrochloric  Acid  and  Leblanc  Soda.  In  two  parts  *i5  oo 

Vol.  III.     Ammonia  Soda *io  oo 

Vol.  IV.  Electrolytic  Methods (In  Press.) 

Technical  Chemists'  Handbook i2mo,  leather,  *3  50 

Technical  Methods  of  Chemical  Analysis.     Trans,  by  C.  A.  Keane. 

in  collaboration  with  the  corps  of  specialists. 

Vol.   I.     In  two  parts Svo,  *i5  oo 

Vol.  n.    In  two  parts Svo,  *i8  oo 

Vol.  Ill (In  Preparation.) 


D.   VAN  NOSTRAND   COMPANY'S   SHORT  TITLE  CATALOG      17 

Lupton,  A.,  Parr,  G.  D.  A.,  and  Perkin,  H.     Electricity  as  Applied  to 

Mining 8vo,  *4  50 

Luquer,  L.  M.     Minerals  in  Rock  Sections 8vo,  *i  50 

Macewen,  H.  A.     Food  Inspection 8vo,  *2  50 

Mackenzie,  N.  F.     Notes  on  Irrigation  Works 8vo,  *2  50 

Mackie,  J.     How  to  Make  a  Woolen  Mill  Pay 8vo,  *2  oo 

Mackrow,  C.     Naval  Architect's  and  Shipbuilder's  Pocket-book. 

i6mo,  leather,  5  oo 

Maguire,  Wm.  R.     Domestic  Sanitary  Drainage  and  Plumbing 8vo,  4  oo 

Mallet,  A.     Compound  Engines.     Trans,  by  R.  R.  Buel.     (Science  Series 

No.  10.) i6mo, 

Mansfield,  A.  N.     Electro-magnets.     (Science  Series  No.  64.) i6rno,  o  50 

Marks,  E.  C.  R.     Construction  of  Cranes  and  Lifting  Machinery.  .  .  .  i2mo,  *i  50 

—  Construction  and  Working  of  Pumps.  .- i2mo,  *i  50 

—  Manufacture  of  Iron  and  Steel  Tubes izmo,  *2  oo 

—  Mechanical  Engineering  Materials I2mo,  *i  oo 

Marks,  G.  C.     Hydraulic  Power  Engineering 8vo,  3  50 

—  Inventions,  Patents  and  Designs I2mo,  *i  oo 

Marlow,  T.  G.     Drying  Machinery  and  Practice 8vo,  *5  oo 

Marsh,  C.  F.     Concise  Treatise  on  Reinforced  Concrete 8vo,  *2  50 

—  Reinforced  Concrete  Compression  Member  Diagram.    Mounted  on 

Cloth  Boards *i  50 

Marsh,  C.  F.,  and  Dunn,  W.     Reinforced  Concrete 4to,  *5  oo 

Marsh,  C.  F.,  and  Dunn,  W.     Manual  of  Reinforced  Concrete  and  Con- 
crete Block  Construction i6mo,  morocco,  *2  50 

Marshall,  W.  J.,  and  Sankey,  H.  R.     Gas  Engines.     (Westminster  Series.) 

8vo,  *2  oo 

Martin.  G,    Triumphs  and  Wonders  of  Modern  Chemistry 8vo,  *2  oo 

Martin,  N.     Properties  and  Design  of  Reinforced  Concrete. 

(In  Press.) 
Massie,  W.  W.,  and  Underbill,  C.  R.     Wireless  Telegraphy  and  Telephony. 

i2mo,  *i  oo 
Matheson,  D.     Australian  Saw-Miller's  Log  and  Timber  Ready  Reckoner. 

i2mo,  leather,  i  50 

Mathot,  R.  E.     Internal  Combustion  Engines 8vo,  *6  oo 

Maurice,  W.     Electric  Blasting  Apparatus  and  Explosives 8vo,  *3  50 

Shot  Firer's  Guide 8vo,  *i  50 

Maxwell,  J.  C.     Matter  and  Motion.     (Science  Series  No.  36.) i6mo,  o  50 

Maxwell,  W.  H.,  and  Brown,  J.  T.     Encyclopedia  of  Municipal  and  Sani- 
tary Engineering 4to,  *io  oo 

Mayer,  A.  M.     Lecture  Notes  on  Physics 8vo,  2  oo 

McCullough,  R.  S.     Mechanical  Theory  of  Heat 8vo,  3  50 

Mclntosh,  J.  G.     Technology  of  Sugar 8vo,  *4  50 

Industrial  Alcohol 8vo,  *3  oo 

Manufacture  of  Varnishes  and  Kindred  Industries.     Three  Volumes. 

8vo. 

Vol.     I.     Oil  Crushing,  Refining  and  Boiling *3  50 

Vol.    II.     Varnish  Materials  and  Oil  Varnish  Making *4  oo 

Vol.  HI.    Spirit  Varnishes  and  Materials *4  50 

McKnight,  J.  D.,  and  Brown,  A.  W.     Marine  Multitubular  Boilers *i  50 


18      D.  VAN   NOSTRAND   COMPANY'S   SHORT  TITLE   CATALOG 

McMaster,  J.  B.     Bridge  and  Tunnel  Centres.     (Science  Series  No.  20.) 

i6mo,  o  50 

McMechen,  F.  L.      Tests  for  Ores,  Minerals  and  Metals i2mo,  *i  oo 

McNeill,  B.     McNeill's  Code 8vo,  *6  oo 

McPherson,  J.  A.     Water-works  Distribution 8vo,  2  50 

Melick,  C.  W.     Dairy  Laboratory  Guide i2mo,  *i  25 

Merck,  E.     Chemical  Reagents;  Their  Purity  and  Tests 8vo,  *i  50 

Merritt,  Wm.  H.     Field  Testing  for  Gold  and  Silver i6mo,  leather,  i  50 

Messer,  W.  A.    Railway  Permanent  Way 8vo,  (In  Press.) 

Meyer,  J.  G.  A.,  and  Pecker,  C.  G.     Mechanical  Drawing  and  Machine 

Design 4to,  5  oo 

Michell,  S.     Mine  Drainage „ 8vo,  10  oo 

Mierzinski,  S.     Waterproofing  of  Fabrics.     Trans,  by  A.  Morris  and  H. 

Robson 8vo,  *2  50 

Miller,  E.  H.     Quantitative  Analysis  for  Mining  Engineers 8vo,  *i  50 

Miller,  G.  A.     Determinants.     (Science  Series  No.  105.) i6mo, 

Milroy,  M.  E.  W.     Home  Lace-making i2mo,  *i  oo 

Minifie,  W.     Mechanical  Drawing 8vo,  *4  oo 

Mitchell,  C.  A.,  and  Prideaux,  R.  M.     Fibres  Used  in  Textile  and  Allied 

Industries 8vo,  *3  oo 

Modern  Meteorology i2mo,  i  50 

Monckton,  C.  C.  F.     Radiotelegraphy.     (Westminster  Series.) 8vo,  *2  oo 

Monteverde,  R.  D.     Vest  Pocket  Glossary  of  English-Spanish,  Spanish- 
English  Technical  Terms 64mo,  leather,  *i  oo 

Moore,  E.  C.  S.     New  Tables  for  the  Complete  Solution  of  Ganguillet  and 

Kutter's  Formula 8vo,  *5  oo 

Morecroft,  J.  H.,  and  Hehre,  F.  W.     Short  Course  in  Electrical  Testing. 

8vo,  *i  50 

Moreing,  C.  A.,  and  Neal,  T.    New  General  and  Mining  Telegraph  Code,  8vo,  *s  oo 

Morgan,  A.  P.     Wireless  Telegraph  Apparatus  for  Amateurs i2mo,  *i  50 

Moses,  A.  J.     The  Characters  of  Crystals 8vo,  *2  oo 

Moses,  A.  J.,  and  Parsons,  C.  L.     Elements  of  Mineralogy. 8vo,  *2  50 

Moss,  S.  A.  Elements  of  Gas  Engine  Design.  (Science  Series  No.i2i.)i6mo,  o  50 

The  Lay-out  of  Corliss  Valve  Gears.   (Science  Series  No.  119.).  i6mo,  o  50 

Mulford,  A.  C.    Boundaries  and  Landmarks (In  Press.) 

Mullin,  J.  P.     Modern  Moulding  and  Pattern- making i2mo,  2  50 

Munby,  A.  E.    Chemistry  and  Physics  of  Building  Materials.     (Westmin- 
ster Series.) 8vo,  *2  oo 

Murphy,  J.  G.     Practical  Mining i6mo,  i  oo 

Murray,  J.  A.     Soils  and  Manures.     (Westminster  Series.) ?,vo,  *2  oo 

Naquet,  A.     Legal  Chemistry I2mo,  2  oo 

Nasmith,  J.     The  Student's  Cotton  Spinning 8vo,  3  oo 

Recent  Cotton  Mill  Construction i2mo,  2  oo 

Neave,  G.  B.,  and  Heilbron,  I.  M.    Identification  of  Organic  Compounds. 

i2mo,  *i  25 

Neilson,  R.  M.     Aeroplane  Patents 8vo,  *2  oo 

Nerz,  F.     Searchlights.     Trans,  by  C.  Rodgers 8vo,  *3  oo 

Nesbit,  A.  F.     Electricity  and  Magnetism (In  Preparation.) 

Neuberger,  H.,  and  Noalhat,  H.     Technology  of  Petroleum.     Trans,  by  J. 

G.  Mclntosh 8vo,  *io  "oo 


D.   VAN   NOSTRAND   COMPANY'S   SHORT  TITLE   CATALOG      19 

Newall,  J.  W.     Drawing,  Sizing  and  Cutting  Bevel-gears 8vo,  i  50 

Nicol,  G.     Ship  Construction  and  Calculations 8vo,  *4  50 

Nipher,  F.  E.     Theory  of  Magnetic  Measurements I2mo,  i  oo 

Nisbet,  H.     Grammar  of  Textile  Design 8vo,  *3  oo 

Nolan,  H.     The  Telescope.     (Science  Series  No.  51.) i6mo,  o  50 

Noll,  A.     How  to  Wire  Buildings I2mo,  i  50 

North,  H.  B.    Laboratory  Notes  of  Experiments  and  General  Chemistry. 

(In  Press.) 

Nugent,  E.     Treatise  on  Optics I2mo,  i  50 

O'Connor,  H.  The  Gas  Engineer's  Pocketbook i2mo,  leather,  3  50 

—  Petrol  Air  Gas i2mo,  *o  75 

Ohm,  G.  S.,  and  Lockwood,  T.  D.  Galvanic  Circuit.  Translated  by 

William  Francis.  (Science  Series  No.  102.) i6mo,  o  50 

Olsen,  J.  C.  Text-book  of  Quantitative  Chemical  Analysis 8vo,  *4  oo 

Olsson,  A.  Motor  Control,  in  Turret  Turning  and  Gun  Elevating.  (U.  S. 

Navy  Electrical  Series,  No.  i.) i2mo,  paper,  *o  50 

Oudin,  M.  A.  Standard  Polyphase  Apparatus  and  Systems 8vo,  *3  oo 

Pakes,  W.  C.  C.,  and  Nankivell,  A.  T.    The  Science  of  Hygiene.  -8vo,  *i  75 

Palaz,  A.     Industrial  Photometry.     Trans,  by  G.  W.  Patterson,  Jr. .  .  8vo,  *4  oo 

Pamely,  C.     Colliery  Manager's  Handbook 8vo,  *io  oo 

Parr,  G.  D.  A.     Electrical  Engineering  Measuring  Instruments 8vo,  *3  50 

Parry,  E.  J.     Chemistry  of  Essential  Oils  and  Artificial  Perfumes.  .  .  .8vo,  *5  oo 

Foods  and  Drugs.     Two  Volumes 8vo, 

Vol.    I.     Chemical  and  Microscopical  Analysis  of  Foods  and  Drugs.  *7  5<> 

Vol.  II.     Sale  of  Food  and  Drugs  Act *3  oo 

Parry,  E.  J.,  and  Coste,  J.  H.     Chemistry  of  Pigments 8vo,  *4  50 

Parry,  L.  A.     Risk  and  Dangers  of  Various  Occupations 8vo,  *3  oo 

Parshall,  H.  F.,  and  Hobart,  H.  M.     Armature  Windings 4to,  *7  50 

Electric  Railway  Engineering 4to,  *io  oo 

Parshall,  H.  F.,  and  Parry,  E.     Electrical  Equipment  of  Tramways. .  .  .  (In  Press.) 

Parsons,  S.  J.     Malleable  Cast  Iron 8vo,  *2  50 

Partington,  J.  R.     Higher  Mathematics  for  Chemical  Students.  .i2mo,  *2  oo 

Passmore,  A.  C.     Technical  Terms  Used  in  Architecture 8vo,  *3  50 

Paterson,  G.  W.  L.     Wiring  Calculations i2mo,  *2  oo 

Patterson,  D.     The  Color  Printing  of  Carpet  Yarns 8vo,  *3  50 

Color  Matching  on  Textiles 8vo,  *3  oo 

The  Science  of  Color  Mixing 8vo,  *3  oo 

Paulding,  C.  P.     Condensation  of  Steam  in  Covered  and  Bare  Pipes. 

8vo,  *2  oo 

Transmission  of  Heat  through  Cold-storage  Insulation i2mo,  *i  oo 

Payne,   D.   W.     Iron  Founders'   Handbook (In   Press.) 

Peddie,  R.  A.     Engineering  and  Metallurgical  Books 12010, 

Peirce,  B.     System  of  Analytic  Mechanics 4to,  10  oo 

Pendred,  V.     The  Railway  Locomotive.     (Westminster  Series.) 8vo,  *2  oo 

Perkin,  F.  M.     Practical  Methods  of  Inorganic  Chemistry i2mo,  *i  oo 

Perrigo,  0.  E.     Change  Gear  Devices 8vo,  i  oo 

Perrine,  F.  A.  C.     Conductors  for  Electrical  Distribution 8vo,  *3  50 

Perry,  J.     Applied  Mechanics : 8vo,  *2  50 

Petit,  G.     White  Lead  and  Zinc  White  Paints 8vo,  *i  50 


20      D.  VAN   NOSTRAND   COMPANY'S  SHORT  TITLE  CATALOG 

Petit,  R.     How  to  Build  an  Aeroplane.     Trans,  by  T.  O'B.  Hubbard,  and 

J.  H.  Ledeboer 8vo,  *i  50 

Pettit,  Lieut.  J.  S.     Graphic  Processes.     (Science  Series  No.  76.) . . .  i6mo,  o  50 
Philbrick,  P.  H.     Beams  and  Girders.     (Science  Series  No.  88.) . . .  i6mo, 

Phillips,  J.     Engineering  Chemistry 8vo,  *4  50 

Gold  Assaying 8vo,  *2  50 

Dangerous  Goods 8vo,  3  50 

Phin,  J.     Seven  Follies  of  Science i2mo,  *i  25 

Pickworth,  C.  N.     The  Indicator  Handbook.     Two  Volumes.  .i2mo,  each,  i  50 

Logarithms  for  Beginners i2mo-  boards,  o  50 

The  Slide  Rule. i2mo,  i  oo 

Plattner's  Manual  of  Blow-pipe  Analysis.    Eighth  Edition,  revised.    Trans. 

by  H.  B.  Cornwall 8vo,  *4  oo 

Plympton,  G.  W.    The  Aneroid  Barometer.    (Science  Series  No.  35.)   i6mo,  o  50 

How  to  become  an  Engineer.     (Science  Series  No.  100.) i6mo,  o  50 

Van  Nostrand's  Table  Book.     (Science  Series  No.  104.) i6mo,  050 

Pochet,  M.  L.     Steam  Injectors.     Translated  from  the  French.     (Science 

Series  No.  29.) i6mo,  o  50 

Pocket  Logarithms  to  Four  Places.     (Science  Series  No.  65.) i6mo,  o  50 

leather,  i  oo 

Polleyn,  F.     Dressings  and  Finishings  for  Textile  Fabrics 8vo,  *3  oo 

Pope,  F.  L.     Modern  Practice  of  the  Electric  Telegraph 8vo,  i  50 

Popplewell,  W.  C.  Elementary  Treatise  on  Heat  and  Heat  Engines .  .  i2mo,  *3  oo 

Prevention  of  Smoke 8vo,  *3  50 

—  Strength  of  Materials 8vo,  *i  75 

Porter,  J.  R.    Helicopter  Flying  Machine i2mo,  *i  25 

Potter,  T.     Concrete 8vo,  *3  oo 

Potts,  H.  E.     Chemistry  of  the  Rubber  Industry.     (Outlines  of  Indus- 
trial Chemistry) 8vo,  *2  oo 

Practical  Compounding  of  Oils,  Tallow  and  Grease ., ............  8vo,  *3  50 

Practical  Iron  Founding i2mo,  i  50 

Pratt,  K.    Boiler  Draught i2mo,  *i  25 

Pray,  T.,  Jr.     Twenty  Years  with  the  Indicator 8vo,  2  50 

Steam  Tables  and  Engine  Constant 8vo,  2  oo 

Calorimeter  Tables 8vo,  i  oo 

Preece,  W.  H.     Electric  Lamps (In  Press.) 

Prelini,  C.     Earth  and  Rock  Excavation 8vo,  *3  oo 

Graphical  Determination  of  Earth  Slopes 8vo,  *2  oo 

Tunneling.    New  Edition 8vo,  *3  oo 

Dredging.    A  Practical  Treatise 8vo,  *3  oo 

Prescott,  A.  B.     Organic  Analysis 8vo,  5  oo 

Prescott,  A.  B.,  and  Johnson,  0.  C.     Qualitative  Chemical  Analysis.  .  .8vo,  *3  50 
Prescott,  A.  B.,  and  Sullivan,  E.  C.     First  Book  in  Qualitative  Chemistry. 

I2mo,  *i  50 

Prideaux,  E.  B.  R.    Problems  in  Physical  Chemistry 8vo,  *2  oo 

Pritchard,  0.  G.     The  Manufacture  of  Electric-light  Carbons .  .  8vo,  paper,  *o  60 
Pullen,  W.  W.  F.     Application  of  Graphic  Methods  to  the  Design  of 

Structures i2mo,  *2  50 

Injectors:  Theory,  Construction  and  Working i2mo,  *i  50 

Pulsifer,  W.  H.     Notes  for  a  History  of  Lead 8vo,  4  oo 

Purchase,  W.  R.     Masonry i2mo,  *3  oo 


D.  VAN   NOSTRAND   COMPANY'S  SHORT  TITLE   CATALOG      21 

Putsch,  A.     Gas  and  Coal-dust  Firing 8vo,  *3  oo 

Pynchon,  T.  R.     Introduction  to  Chemical  Physics 8vo,  3  oo 

Rafter  G.  W.     Mechanics  of  Ventilation.     (Science  Series  No.  33.) .  i6mo,  o  50 

Potable  Water,     (Science  Series'No.  103.) i6mc  50 

Treatment  of  Septic  Sewage.     (Science  Series  No.  118.). . . .  i6mo  50 

Rafter,  G.  W.,  and  Baker,  M.  N.     Sewage  Disposal  in  the  United  States, 

4to,  *6  oo 

Raikes,  H.  P.     Sewage  Disposal  Works 8vo,  *4  oo 

Railway  Shop  Up-to-Date 4to,  2  oo 

Ramp,  H.  M.     Foundry  Practice (In  Press.) 

Randall,  P.  M.     Quartz  Operator's  Handbook I2mo,  2  oo 

Randau,  P.     Enamels  and  Enamelling 8vo,  *4  oo 

Rankine.  W.  J.  M.     Applied  Mechanics 8vo,  5  oo 

Civil  Engineering 8vo,  6  50 

Machinery  and  Millwork 8vo,  5  oo 

—  The  Steam-engine  and  Other  Prime  Movers 8vo,  5  oo 

Useful  Rules  and  Tables 8vo,  4  oo 

Rankine,  W.  J.  M.,  and  Bamber,  E.  F.     A  Mechanical  Text-book. . .  .8vo,  3  50 
Raphael,  F.  C.     Localization  of  Faults  in  Electric  Light  and  Power  Mains. 

8vo,  *3  oo 

Rasch,  E.    Electric  Arc.    Trans,  by  K.  Toraberg (In  Press.) 

Rathbone,  R.  L.  B.     Simple  Jewellery 8vo,  *2  oo 

Rateau,  A.     Flow  of  Steam  through  Nozzles  and  Orifices.     Trans,  by  H. 

B.  Brydon 8vo,  *i  50 

Rausenberger,  F.     The  Theory  of  the  Recoil  of  Guns 8vo,  *4  50 

Rautenstrauch,  W.    Notes  on  the  Elements  of  Machine  Design .  8 vo,  boards,  * i  50 
Rautenstrauch,  W.,  and  Williams,  J.  T.     Machine  Drafting  and  Empirical 

Design. 

Part   I.  Machine  Drafting 8vo,  *i  25 

Part  II.  Empirical  Design (In  Preparation.) 

Raymond,  E.  B.     Alternating  Current  Engineering i2mo,  *2  50 

Rayner,  H.     Silk  Throwing  and  Waste  Silk  Spinning 8vo,  *2  50 

Recipes  for  the  Color,  Paint,  Varnish,  Oil,  Soap  and  Drysaltery  Trades .  8vo,  *3  50 

Recipes  for  Flint  Glass  Making i2mo,  *4  50 

Redfern,  J.  B.    Bells,  Telephones  (Installation  Manuals  Series)  i6mo, 

(In  Press.) 

Redwood,  B.     Petroleum.     (Science  Series  No.  92.) i6mo,  o  50 

Reed's  Engineers'  Handbook 8vo,  *5  oo 

Key  to  the  Nineteenth  Edition  of  Reed's  Engineers'  Handbook . .  8vo,  *3  oo 

Useful  Hints  to  Sea-going  Engineers i2mo,  i  50 

Marine  Boilers I2mo,  2  oo 

Reinhardt,  C.  W.     Lettering  for  Draftsmen,  Engineers,  and  Students. 

oblong  4to,  boards,  i  oo 

The  Technic  of  Mechanical  Drafting oblong  4to,  boards,  *i  oo 

Reiser,  F.     Hardening  and  Tempering  of  Steel.     Trans,  by  A.  Morris  and 

H.  Robson i2mo,  *2  50 

Reiser,  N.     Faults  in  the  Manufacture  of  Woolen  Goods.     Trans,  by  A. 

Morris  and  H.  Robson 8vo,  *2  50 

Spinning  and  Weaving  Calculations 8vo,  *5  oo 

Renwick,  W.  G.     Marble  and  Marble  Working 8vo,  5  oo 


22      D.   VAN    NOSTRAND   COMPANY'S  SHORT  TITLE   CATALOG 

Reynolds,   0.,   and  Idell,   F.   E.     Triple   Expansion  Engines.     (Science 

Series  No.  99.) i6mo,  o  50- 

Rhead,  G.  F.     Simple  Structural  Woodwork i2mo,  *i  oo 

Rice,  J.  M.,  and  Johnson,  W.  W.     A  New  Method  of  Obtaining  the  Differ- 
ential of  Functions i2mo,  o  50 

Richards,  W.  A.  and  North,  H.  B.     Manual  of  Cement  Testing.  (In  Press.) 

Richardson,  J.     The  Modern  Steam  Engine 8vo,  *3  50 

Richardson,  S.  S.     Magnetism  and  Electricity i2mo,  *2  oo 

Rideal,  S.     Glue  and  Glue  Testing 8vo,  *4  oo 

Rings,  F.     Concrete  in  Theory  and  Practice i2mo,  *2  50 

Ripper,  W.     Course  of  Instruction  in  Machine  Drawing folio,  *6  oo 

Roberts,  F.  C.     Figure  of  the  Earth.     (Science  Series  No.  79.) i6mo,  o  50 

Roberts,  J.,  Jr.     Laboratory  Work  in  Electrical  Engineering 8vo,  *2  oo 

Robertson,  L.  S.     Water-tube  Boilers 8vo,  3  oo 

Robinson,  J.  B.     Architectural  Composition 8vo,  *2  50 

Robinson,  S.  W.     Practical  Treatise  on  the  Teeth  of  Wheels.     (Science 

Series  No.  24.) i6mo,  o  50 

Railroad  Economics.     (Science  Series  No.  59.) i6mo,  o  50 

Wrought  Iron  Bridge  Members.     (Science  Series  No.  60.) i6mo,  o  50 

Robson,  J.  H.     Machine  Drawing  and  Sketching 8vo,  *i  50 

Roebling,  J   A.     Long  and  Short  Span  Railway  Bridges folio,  25  oo 

Rogers,  A.     A  Laboratory  Guide  of  Industrial  Chemistry i2mo,  *i  50 

Rogers,  A.,  and  Aubert,  A.  B.     Industrial  Chemistry 8vo,  *5  oo 

Rogers,  F.     Magnetism  of  Iron  Vessels.     (Science  Series  No.  30.) . .  i6mo,  o  50 

Rohland,  P.     Colloidal  and  Cyrstalloidal   State  of  Matter.     Trans,  by  j 

.  W.  J.  Britland  and  H.  E.  Potts i2mo,  *i  25 

Rollins,  W.     Notes  on  X-Light 8vo,  *$  oo 

Rollinson,  C.    Alphabets Oblong,  .i2mo,  (In  Press.) 

Rose,  J.     The  Pattern-makers'  Assistant 8vo,  2  50 

Key  to  Engines  and  Engine-running i2mo,  2  50 

Rose,  T.  K.     The  Precious  Metals.     (Westminster  Series.) 8vo,  *2  oo 

Rosenhain,  W.     Glass  Manufacture.     (Westminster  Series.) 8vo,  *2  oo 

Ross,  W.  A.     Plowpipe  in  Chemistry  and  Metallurgy i2mo,  *2  oo 

Rossiter,  J.  T.     Steam  Engines.     (Westminster  Series.). .  .  .8vo  (In  Press.) 

Pumps  and  Pumping  Machinery.     (Westminster  Series.).. 8vo  (In  Press.) 

Roth.     Physical  Chemistry 8vo,  *2  oo 

Rouillion,  L.     The  Economics  of  Manual  Training 8vo,  2  oo 

Rowan,  F.  J.     Practical  Physics  of  the  Modern  Steam-boiler 8vo,  7  50 

Rowan,   F.   J.,   and  Idell,   F.   E.     Boiler  Incrustation  and  Corrosion. 

(Science  Series  No.  27.) i6mo,  o  50 

Roxburgh,  W.     General  Foundry  Practice 8vo,  *3  50 

Ruhmer,  E.     Wireless  Telephony.     Trans,  by  J.  Erskine-Murray .  .  .  .8vo,  *3  50 

Russell,  A.     Theory  of  Electric  Cables  and  Networks 8vo,  *3  oo 

Sabine,  R.     History  and  Progress  of  the  Electric  Telegraph i2mo,  i  25 

Saeltzer  A.     Treatise  on  Acoustics I2mo,  i  oo 

Salomons,  D.     Electric  Light  Installations.     i2mo. 

Vol.    I.     The  Management  of  Accumulators 2  50 

Vol.  II.     Apparatus , .  2  25 

Vol.  III.     Applications i  5<> 

Sanford,  P.  G.     Nitro-explosives 8vo,  *4  oo 


D.  VAN   NOSTRAND   COMPANY'S    SHORT   TITLE  CATALOG      23 

Saunders,  C.  H.     Handbook  of  Practical  Mechanics i6mo,  i  oo 

leather,  i  25 

Saunnier,  C.     Watchmaker's  Handbook i2mo,  3  oo 

Sayers,  H.  M.     Brakes  for  Tram  Cars 8vo,  *i  25 

Scheele,  C.  W.     Chemical  Essays 8vo,  *2  oo 

Schellen,  H.     Magneto-electric  and  Dynamo-electric  Machines 8vo,  5  oo 

Scherer,  R.     Casein.     Trans,  by  C.  Salter 8vo,  *3  oo 

Schidrowitz,  P.    Rubber,  Its  Production  and  Industrial  Uses 8vo,  *5  oo 

Schindler,  K.     Iron  and  Steel  Construction  Works. 

Schmall,  C.  N.     First  Course  in  Analytic  Geometry,  Plane  and  Solid. 

i2mo,  half  leather,  *i  75 

Schmall,  C.  N.,  and  Shack,  S.  M.     Elements  of  Plane  Geometry i2mo,  *i  25 

Schmeer,  L.     Flow  of  Water 8vo,  *3  oo 

Schumann,  F.     A  Manual  of  Heating  and  Ventilation i2mo,  leather,  i  50 

Schwarz,  E.  H.  L.     Causal  Geology 8vo,  *2  50 

Schweizer,  V.,  Distillation  of  Resins 8vo,  *3  50 

Scott,  W.  W.     Qualitative  Analysis.     A  Laboratory  Manual 8vo,  *i  50 

Scribner,  J.  M.     Engineers'  and  Mechanics'  Companion  .  . .  i6mo,  leather,  i  50 

Searle,  A.  B.     Modern  Brickmaking 8vo,  *5  oo 

Searle,  G.  M.     "  Sumners'  Method."     Condensed  and  Improved.    (Science 

Series  No.  124.) i6mo,  o  50 

Seaton,  A.  E.     Manual  of  Marine  Engineering 8vo,  6  oo 

Seaton,  A.  E.,  and  Rounthwaite,  H.  M.     Pocket-book  of  Marine  Engineer- 
ing  i6mo,  leather,  3  oo 

Seeligmann,  T.,  Torrilhon,  G.  L.,  and  Falconnet,  H.     India  Rubber  and 

Gutta  Percha.     Trans,  by  J.  G.  Mclntosh 8vo,  *s  oo 

Seidell,  A.     Solubilities  of  Inorganic  and  Organic  Substances 8vo,  *3  oo 

Sellew,  W.  H.     Steel  Rails 410  (In  Press.) 

Senter,  G.     Outlines  of  Physical  Chemistry I2mo,  *i  75 

Textbook  of  Inorganic  Chemistry i2mo,  *i  75 

Sever,  G.  F.     Electric  Engineering  Experiments 8vo,  boards,  *i  oo 

Sever,  G.  F.,  and  Townsend,  F.     Laboratory  and  Factory  Tests  in  Electrical, 

Engineering 8vo,  *2  50 

Sewall,  C.  H.     Wireless  Telegraphy 8vo,  *2  oo 

Lessons  in  Telegraphy i2mo,  *i  oo 

Sewell,  T.     Elements  of  Electrical  Engineering .  .8vo,  *3  oo 

The  Construction  of  Dynamos 8mo,  *3  oo 

Sexton,  A.  H.     Fuel  and  Refractory  Materials i2mo,  *2  50 

Chemistry  of  the  Materials  of  Engineering I2mo,  *2  50 

Alloys  (Non-Ferrous) 8vo,  *3  oo 

The  Metallurgy  of  Iron  and  Steel 8vo,  *6  50 

Seymour,  A.     Practical  Lithography 8vo,  *2  50 

Modern  Printing  Inks 8vo,  *2  oo 

Shaw,  Henry  S.  H.     Mechanical  Integrators.     (Science  Series  No.  83.) 

i6mo,  o  50 

Shaw,  P.  E.     Course  of  Practical  Magnetism  and  Electricity 8vo,  *i  oo 

Shaw,  S.     History  of  the  Staffordshire  Potteries 8vo,  *3  oo 

Chemistry  of  Compounds  Used  in  Porcelain  Manufacture 8vo,  *5  oo 

Shaw,  W.  N.    Forecasting  Weather 8vo,  *3  50 

Sheldon,  S.,  and  Hausmann,  E.    Direct  Current  Machines i2mo,  *2  50 

Alternating  Current  Machines i2mo,  *2  50 


24     D.  VAN   NOSTRAND   COMPANY'S   SHORT  TITLE   CATALOG 

Sheldon,  S.,  and  Hausmann,  E.     Electric  Traction  and  Transmission 

Engineering i2mo,  *2  50 

Sherriff,  F.  F.     Oil  Merchants'  Manual i2mo,  *3  50 

Shields,  J.  E.     Notes  on  Engineering  Construction i2mo,  i  50 

Shock,  W.  H.     Steam  Boilers 4to,  half  morocco,  15  oo 

Shreve,  S.  H.     Strength  of  Bridges  and  Roofs 8vo,  3  50 

Shunk,  W.  F.     The  Field  Engineer i2mo,  morocco,  2  50 

Simmons,  W.  H.,  and  Appleton,  H.  A.    Handbook  of  Soap  Manufacture. 

8vo,  *3  oo 

Simmons,  W.  H.,  and  Mitchell,  C.  A.     Edible  Fats  and  Oils 8vo,  *3  oo 

Simms,  F.  W.     The  Principles  and  Practice  of  Leveling 8vo,  2  50 

Practical  Tunneling 8vo,  7  50 

Simpson,  G.    The  Naval  Constructor i2mo,  morocco,  *s  oo 

Simpson,   W.     Foundations 8vo,    (In   Press.) 

Sinclair,  A.     Development  of  the  Locomotive  Engine  .  . .  8vo,  half  leather,  5  oo 

Sinclair,  A.     Twentieth  Century  Locomotive 8vo,  half  leather,  *s  oo 

Sindall,  R.  W.     Manufacture  of  Paper.     (Westminster  Series.) 8vo,  *2  oo 

Sloane,  T.  O'C.     Elementary  Electrical  Calculations i2mo,  *2  oo 

Smith,  C.  A.  M.     Handbook  of  Testing,  MATERIALS 8vo,  *2  50 

Smith,  C.  A.  M.,  and  Warren,  A.  G.     New  Steam  Tables 8vo, 

Smith,  C.  F.     Practical  Alternating  Currents  and  Testing 8vo,  *2  50 

Practical  Testing  of  Dynamos  and  Motors 8vo,  *2  oo 

Smith,  F.  E.     Handbook  of  General  Instruction  for  Mechanics.  .  . .  i2mo,  i  50 

Smith,  J.  C.     Manufacture  of  Paint 8vo,  *3  oo 

Smith,  R.  H.    Principles  of  Machine  Work i2mo,  *3  j>o 

Elements  of  Machine  Work i2mo,  *2  oo 

Smith,  W.     Chemistry  of  Hat  Manufacturing i2mo,  *3  oo 

Snell,  A.  T.     Electric  Motive  Power 8vo,  *4  oo 

Snow,  W.  G.     Pocketbook  of  Steam  Heating  and  Ventilation.    (In  Press.) 
Snow,  W.  G.,  and  Nolan,  T.     Ventilation  of  Buildings.     (Science  Series 

No.  5.) i6mo,  o  50 

Soddy,  F.     Radioactivity. 8vo,  *3  oo 

Solomon,  M.     Electric  Lamps.     (Westminster  Series.) 8vo,  *2  oo 

Sothern,  J.  W.     The  Marine  Steam  Turbine 8vo,  *5  oo 

Southcombe,  J.  E.    Faults,  Oils  and  Varnishes.     (Outlines  of  Indus- 
trial Chemistry.) 8vo,   (In  Press.) 

Soxhlet,  D.  H.     Dyeing  and  Staining  Marble.     Trans,  by  A.  Morris  and 

H.  Robson 8vo,  *2  50 

Spang,  H.  W.     A  Practical  Treatise  on  Lightning  Protection i2mo,  i  oo 

Spangenburg,    L.     Fatigue    of    Metals.     Translated   by    S.    H.    Shreve. 

(Science  Series  No.  23.) i6mo,  o  50 

Specht,  G.  J.,  Hardy,  A.  S.,  McMaster,  J.B  .,  and  Walling.     Topographical 

Surveying.     (Science  Series  No.  72.) i6mo,  o  50 

Speyers,  C.  L.     Text-book  of  Physical  Chemistry 8vo,  *2  25 

Stahl,  A.  W.     Transmission  of  Power.     (Science  Series  No.  28.) . . .  i6mo, 

Stahl,  A.  W.,  and  Woods,  A.  T.     Elementary  Mechanism I2mo,  *2  oo 

Staley,  C.,  and  Pierson,  G.  S.     The  Separate  System  of  Sewerage 8vo,  *3  oo 

Standage,  H.  C.     Leatherworkers'  Manual 8vo,  *3  50 

Sealing  Waxes,  Wafers,  and  Other  Adhesives 8vo,  *2  oo 

Agglutinants  of  all  Kinds  for  all  Purposes I2mo,  *3  So 

Stansbie,  J.  H.     Iron  and  Steel.     (Westminster  Series.) 8vo,  *2  oo 


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Steinman,  D.  B.     Suspension  Bridges  and  Cantilevers.     (Science  Series 

No.  127) o  50 

Stevens,  H.  P.     Paper  Mill  Chemist i6mo,  *2  50 

Stevenson,  J.  L.     Blast-Furnace  Calculations I2mo,  leather,  *2  oo 

Stewart,  A.     Modern  Polyphase  Machinery i2mo,  *2  oo 

Stewart,  G.     Modern  Steam  Traps i2mo,  *i  25 

Stiles,  A.     Tables  for  Field  Engineers i2mo,  i  oo 

Stillman,  P.     Steam-engine  Indicator i2mo,  i  oo 

Stodola,  A.     Steam  Turbines.     Trans,  by  L.  C.  Loewenstein 8vo,  *s  oo 

Stone,  H.     The  Timbers  of  Commerce 8vo,  3  50 

Stone,  Gen.  R.    New  Roads  and  Road  Laws i2mo,  i  oo 

Stopes,  M.     Ancient  Plants 8vo,  *2  oo 

The  Study  of  Plant  Life 8vo,  *2  oo 

Stumpf ,  Prof.    Una-Flow  of  Steam  Engine (In  Press.} 

Sudborough,  J.  J.,  and  James,  T.  C.     Practical  Organic  Chemistry.  .  i2mo,  *2  oo 

Suffling,  E.  R.     Treatise  on  the  Art  of  Glass  Painting 8vo,  *3  50 

Swan,  K.     Patents,  Designs  and  Trade  Marks.      (Westminster  Series.). 8 vo,  *2  oo 

Sweet,  S.  H.     Special  Report  on  Coal 8vo,  3  oo 

Swinburne,  J.,  Wordingham,  C.  H.,  and  Martin,  T.  C.     Eletcric  Currents. 

(Science  Series  No.  109.) i6mo,  o  50 

Swoope,  C.  W.     Practical  Lessons  in  Electricity i2mo,  *2  oo 

Tailfer,  L.     Bleaching  Linen  and  Cotton  Yarn  and  Fabrics 8vo,  *5  oo 

Tate,  J.  S.     Surcharged  and  Different  Forms  of  Retaining-walls.     (Science 

Series  No.  7.) i6mo,  o  50 

Taylor,  E.  N.     Small  Water  Supplies i2mo,  *2  oo 

Templeton,  W.     Practical  Mechanic's  Workshop  Companion. 

i2mo,  morocco,  2  oo 
Terry,  H.  L.     India  Rubber  and  its  Manufacture.     (Westminster  Series.) 

8vo,  *2  oo 
Thayer,  H.  R.     Structural  Design.    8vo. 

Vol.     I.    Elements  of  Structural  Design *2  oo 

Vol.    II.    Design  of  Simple  Structures (In  Preparation.} 

Vol.  III.    Design  of  Advanced  Structures (In  Preparation.} 

Thiess,  J.  B.  and  Joy,  G.  A.    Toll  Telephone  Practice 8vo,  *3  50 

Thorn,  C.,  and  Jones,  W.  H.     Telegraphic  Connections oblong  i2mo,  i  50 

Thomas,  C.  W.     Paper-makers'  Handbook (In  Press.} 

Thompson,  A.  B.     Oil  Fields  of  Russia 4to,  *7  50 

Petroleum  Mining  and  Oil  Field  Development 8vo,  *5  oo 

Thompson,  E.  P.     How  to  Make  Inventions 8vo,  o  50 

Thompson,  S.  P.     Dynamo  Electric  Machines.     (Science  Series  No.  75.) 

i6mo,  o  50 

Thompson,  W.  P.     Handbook  of  Patent  Law  of  All  Countries i6mo,  i  50 

Thomson,  G.  S.    Milk  and  Cream  Testing 121110,  *i  75 

Modern  Sanitary  Engineering,  House  Drainage,  etc.  8vo,  (In  Press.} 

Thornley,  T.     Cotton  Combing  Machines 8vo,  *3  oo 

Cotton  Spinning.     8vo. 

First  Year *i  50 

Second  Year *a  50 

Third  Year *2  50 

Thurso,  J.  W.     Modern  Turbine  Practice 8vo,  *4  oo 


26     D.   VAN   NOSTRAND    COMPANY'S   SHORT  TITLE  CATALOG 

Tidy,  C.  Meymott.     Treatment  of  Sewage.     (Science  Series  No.   94.). 

i6mo,  o  50 

Tinney,  W.  H.     Gold-mining  Machinery 8vo,  *3  oo 

Titherley,  A.  W.     Laboratory  Course  of  Organic  Chemistry 8vo,  *2  oo 

Toch,  M.     Chemistry  and  Technology  of  Mixed  Paints 8vo,  *3  oo 

Materials  for  Permanent  Painting i2mo,  *2  oo 

Todd,  J.,  and  Whall,  W.  B.     Practical  Seamanship 8vo,  *7  50 

Tonge,  J.     Coal.     (Westminster  Series.) 8vo,  *2  oo 

Townsend,  F.     Alternating  Current  Engineering 8vo,  boards  *o  75 

Townsend,  J.     lonization  of  Gases  by  Collision. 8vo,  *i  25 

Transactions  of  the  Amerkan  Institute  of  Chemical  Engineers.     8vo. 

Vol.      I.     1908 *6  oo 

Vol.    II.     1909 *6  oo 

Vol.  HI.      1910 *6  oo 

Vol.  IV.     1911 *6  oo 

Traverse  Tables.     (Science  Series  No.  115.) i6mo,  o  50 

morocco,  i  oo 
Trinks,  W.,  and  Housum,  C.     Shaft  Governors.     (Science  Series  No.  122.) 

i6mo,  o  50 

Trowbridge,  W.  P.     Turbine  Wheels.     (Science  Series  No.  44.) i6mo,  o  50 

Tucker,  J.  H.     A  Manual  of  Sugar  Analysis 8vo,  3  50 

Tumlirz,  0.     Potential.     Trans,  by  D.  Robertson i2mo,  i  25 

Tunner,  P.  A.     Treatise  on  Roll-turning.     Trans,  by  J.  B.  Pearse. 

8vo,  text  and  folio  atlas,  10  oo 

Turbayne,  A.  A.     Alphabets  and  Numerals 4to,  2  oo 

Turnbull,  Jr.,  J.,  and  Robinson,  S.  W.     A  Treatise  on  the  Compound 

Steam-engine,      (Science  Series  No.  8.) i6mo, 

Turrill,  S.  M.     Elementary  Course  in  Perspective i2mo,  *i  25 

Underbill,  C.  R.     Solenoids,  Electromagnets  and  Electromagnetic  Wind- 
ings  I2mo,  *2  oo 

Universal  Telegraph  Cipher  Code i2mo,  i  oo 

Urquhart,  J.  W.     Electric  Light  Fitting i2mo,  2  oo 

Electro-plating i2mo,  2  oo 

Electrotyping i2mo,  2  oo 

Electric  Ship  Lighting i2mo,  3  oo 

Vacher,  F.  Food  Inspector's  Handbook i2mo,  *2  50 

Van  Nostrand's  Chemical  Annual.  Second  issue  1909 i2mo,  *2  50 

Year  Book  of  Mechanical  Engineering  Data.  First  issue  1912 .  . .  (In  Press.) 

Van  Wagenen,  T.  F.  Manual  of  Hydraulic  Mining i6mo,  i  oo 

Vega,  Baron  Von.  Logarithmic  Tables 8vo,  half  morocco,  2  oo 

Villon,  A.  M.  Practical  Treatise  on  the  Leather  Industry.  Trans,  by  F. 

T.  Addyman 8vo,  *io  oo 

Vincent,  C.  Ammonia  and  its  Compounds.  Trans,  by  M.  J.  Salter .  .  8vo,  *2  oo 

Volk,  C.  Haulage  and  Winding  Appliances 8vo,  *4  oo 

Von  Georgievics,  G.  Chemical  Technology  of  Textile  Fibres.  Trans,  by 

C.  Salter 8vo,  *4  50 

Chemistry  of  Dyestuffs.  Trans,  by  C.  Salter 8vo,  *4  50 

Vose,  G.  L.  Graphic  Method  for  Solving  Certain  Questions  in  Arithmetic 

and  Algebra.     (Science  Series  No.  16.) i6mo,  o  50 


D.   VAN  NOSTRAND  COMPANY'S    SHORT  TITLE  CATALOG    27 

Wabner,  R.     Ventilation  in  Mines.     Trans,  by  C.  Salter .8vo,  *4  50 

Wade,  E.  J.     Secondary  Batteries 8vo,  *4  oo 

Wadmore,  T.  M.    Elementary  Chemical  Theory.. i2mo,  *i  50 

Wadsworth,  C.     Primary  Battery  Ignition i2mo  (In  Press.) 

Wagner,  E.     Preserving  Fruits,  Vegetables,  and  Meat i2mo,  *2  50 

Waldram,  P.  J.      Principles  of  Structural  Mechanics (In  Press.) 

Walker,  F.     Aerial  Navigation 8vo,  2  oo 

—  Dynamo  Building.     (Science  Series  No.  98.) i6mo,  o  50 

Electric  Lighting  for  Marine  Engineers 8vo,  2  oo 

Walker,  S.  F.     Steam  Boilers,  Engines  and  Turbines 8vo,  3  oo 

Refrigeration,  Heating  and  Ventilation  on  Shipboard i2mo,  *2  oo 

Electricity  in  Mining 8vo,  *3  50 

Walker,  W.  H.     Screw  Propulsion 8vo,  o  75 

Wallis-Tayler,  A.  J.     Bearings  and  Lubrication 8vo,  *i  50 

Aerial  or  Wire  Ropeways 8vo,  *3  oo 

Modern  Cycles 8vo,  4  oo 

Motor  Cars 8vo,  i  80 

Motor  Vehicles  for  Business  Purposes 8vo,  3  50 

Pocket  Book  of  Refrigeration  and  Ice  Making i2mo,  i  50 

Refrigeration,  Cold  Storage  and  Ice-Making 8vo,  *4  50 

Sugar  Machinery I2mo,  *2  oo 

Wanklyn,  J.  A.     Water  Analysis I2mo,  2  oo 

Wansbrough,  W.  D.     The  A  B  C  of  the  Differential  Calculus 12 mo,  *i  50 

Slide  Valves i2mo,  *2  oo 

Ward,  J.  H.     Steam  for  the  Million 8vo,  -i  oo 

Waring,  Jr.,  G.  E.     Sanitary  Conditions.     (Science  Series  No.  31.).  .i6mo,  050 

Sewerage  and  Land  Drainage *6  oo 

Waring,  Jr.,  G.  E.     Modern  Methods  of  Sewage  Disposal i2mo,  2  oo 

• How  to  Drain  a  House I2mo,  i  25 

Warren,  F.  D.     Handbook  on  Reinforced  Concrete I2mo,  *2  50 

Watkins,  A.    Photography.     (Westminster  Series.) 8vo,  *2  oo 

Watson,  E.  P.     Small  Engines  and  Boilers I2mo,  i  25 

Watt,  A.     Electro-plating  and  Electro-refining  of  Metals 8vo,  *4  50 

Electro-metallurgy i2mo,  i  oo 

The  Art  of  Soap-making 8vo,  3  oo 

Leather  Manufacture 8vo,  *4  oo 

Paper-Making 8vo,  3  oo 

Weale,  J.     Dictionary  of  Terms  Used  in  Architecture i2mo,  2  50 

Weale's  Scientific  and  Technical  Series.     (Complete  list  sent  on  applica- 
tion.) 

Weather  and  Weather  Instruments i2mo,  i  oo 

paper,  o  50 

Webb,  H.  L.     Guide  to  the  Testing  of  Insulated  Wires  and  Cables. .  izmo,  i  oo 

Webber,  W.  H.  Y.     Town  Gas.     (Westminster  Series.) 8vo,  *2  oo 

Weisbach,  J.     A  Manual  of  Theoretical  Mechanics 8vo,  *6  oo 

sheep,  *7  50 

Weisbach,  J.,  and  Herrmann,  G.     Mechanics  of  Air  Machinery 8vo,  *3  75 

Welch,  W.     Correct  Lettering (In  Press.) 

Weston,  E.  B.     Loss  of  Head  Due  to  Friction  of  Water  in  Pipes  . . .  I2mo,  *i .  50 

Weymouth,  F.  M.     Drum  Armatures  and  Commutators 8vo,  *3  oo 

Wheatley,  O.    Ornamental  Cement  Work (In  Press.) 


28     D.  VAN  NOSTRAND  COMPANY'S  SHORT  TITLE  CATALOG 

Wheeler,  J.  Bf     Art  of  War i2mo,  i  75 

Field  Fortifications i2mo,  i  75 

Whipple,  S.     An  Elementary  and  Practical  Treatise  on  Bridge  Building. 

8vo,  3  oo 

Whithard,  P.     Illuminating  and  Missal  Painting i2mo,  i  50 

Wilcox,  R.  M.     Cantilever  Bridges.     (Science  Series  No.  25.) i6mo,  o  50 

Wilkinson,  H.  D.     Submarine  Cable  Laying  and  Repairing 8vo,  *6  oo 

Williams,  A.  D.,  Jr.,  and  Hutchinson,  R.  W.     The  Steam  Turbine (In  Press.) 

Williamson,  J.,  and  Blackadder,  H.  Surveying 8vo,   (In  Press.) 

Williamson,  R.  S.     On  the  Use  of  the  Barometer 4to,  15  oo 

Practical  Tables  in  Meteorology  and  Hypsometery 4to,  2  50 

Willson,  F.  N.     Theoretical  and  Practical  Graphics 4to,  *4  oo 

Wimperis,  H.  E.     Internal  Combustion  Engine 8vo,  *3  oo 

Winchell,  N.  H.,  and  A.  N.     Elements  of  Optical  Mineralogy 8vo,  *3  50 

Winkler,  C.,  and  Lunge,  G.     Handbook  of  Technical  Gas- Analysis. .  .8vo,  4  oo 

Winslow,  A.     Stadia  Surveying.     (Science  Series  No.  77.) i6mo,  o  50 

Wisser,   Lieut.   J.   P.     Explosive  Materials.     (Science   Series  No.   70.). 

i6mo,  o  50 

Wisser,  Lieut.  J.  P.     Modern  Gun  Cotton.     (Science  Series  No.  89.)  i6mo,  050 

Wood,  De  V.     Luminiferous  Aether.     (Science  Series  No.  85.) ....  i6mo,  o  50 
Woodbury,  D.  V.     Elements  of  Stability  in  the  Well-proportioned  Arch. 

8vo,  half  morocco,  4  oo 

Worden,  E.  C.     The  Nitrocellulose  Industry.     Two  Volumes 8vo,  *io  oo 

Cellulose  Acetate 8vo,  (In  Press.) 

Wright,  A.  C.     Analysis  of  Oils  and  Allied  Substances 8vo,  *3  50 

Simple  Method  for  Testing  Painters'  Materials 8vo,  *2  50 

Wright,  F.  W.     Design  of  a  Condensing  Plant I2mo,  *i  50 

Wright,  H.  E.     Handy  Book  for  Brewers 8vo,  *5  oo 

Wright,  J.    Testing,  Fault  Finding,  etc.,  for  Wiremen.      (Installation 

Manuals  Series.) i6mo,  *o  50 

Wright,  T.  W.     Elements  of  Mechanics 8vo,  *2  50 

Wright,  T.  W.,  and  Hayford,  J.  F.     Adjustment  of  Observations 8vo,  *3  oo 

Young,  J.  E.     Electrical  Testing  for  Telegraph  Engineers 8vo,  *4  oo 

Zahner,  R.     Transmission  of  Power.     (Science  Series  No.  40.) ....  i6mo, 

Zeidler,  J.,  and  Lustgarten,  J.     Electric  Arc  Lamps 8vo,  *2  oo 

Zeuner,  A.     Technical  Thermodynamics.     Trans,  by  J.  F.  Klein.     Two 

Volumes 8vo,  *8  oo 

Zimmer,  G.  F.     Mechanical  Handling  of  Material 4to,  *io  oo 

Zipser,  J.    Textile  Raw  Materials.     Trans,  by  C.  Salter 8vo,  *5  oo 

Ziit  Nedden,  F.    Engineering  Workshop  Machines  and  Processes.     Trans. 

by  J.  A.  Davenport 8 vo  *2  oo 


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